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Theorem 2ndcctbss 17471
Description: If a topology is second-countable, every base has a countable subset which is a base. Exercise 16B2 in Willard. (Contributed by Jeff Hankins, 28-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
Hypotheses
Ref Expression
2ndcctbss.1  |-  X  = 
U. B
2ndcctbss.2  |-  J  =  ( topGen `  B )
2ndcctbss.3  |-  S  =  { <. u ,  v
>.  |  ( u  e.  c  /\  v  e.  c  /\  E. w  e.  B  ( u  C_  w  /\  w  C_  v ) ) }
Assertion
Ref Expression
2ndcctbss  |-  ( ( B  e.  TopBases  /\  J  e.  2ndc )  ->  E. b  e. 
TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  ( topGen `  b
) ) )
Distinct variable groups:    b, c, u, v, w, B    J, b, c
Allowed substitution hints:    S( w, v, u, b, c)    J( w, v, u)    X( w, v, u, b, c)

Proof of Theorem 2ndcctbss
Dummy variables  d 
f  m  n  o  t  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 448 . . 3  |-  ( ( B  e.  TopBases  /\  J  e.  2ndc )  ->  J  e.  2ndc )
2 is2ndc 17462 . . 3  |-  ( J  e.  2ndc  <->  E. c  e.  TopBases  ( c  ~<_  om  /\  ( topGen `
 c )  =  J ) )
31, 2sylib 189 . 2  |-  ( ( B  e.  TopBases  /\  J  e.  2ndc )  ->  E. c  e. 
TopBases  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) )
4 vex 2919 . . . . . . 7  |-  c  e. 
_V
54, 4xpex 4949 . . . . . 6  |-  ( c  X.  c )  e. 
_V
6 3simpa 954 . . . . . . . 8  |-  ( ( u  e.  c  /\  v  e.  c  /\  E. w  e.  B  ( u  C_  w  /\  w  C_  v ) )  ->  ( u  e.  c  /\  v  e.  c ) )
76ssopab2i 4442 . . . . . . 7  |-  { <. u ,  v >.  |  ( u  e.  c  /\  v  e.  c  /\  E. w  e.  B  ( u  C_  w  /\  w  C_  v ) ) }  C_  { <. u ,  v >.  |  ( u  e.  c  /\  v  e.  c ) }
8 2ndcctbss.3 . . . . . . 7  |-  S  =  { <. u ,  v
>.  |  ( u  e.  c  /\  v  e.  c  /\  E. w  e.  B  ( u  C_  w  /\  w  C_  v ) ) }
9 df-xp 4843 . . . . . . 7  |-  ( c  X.  c )  =  { <. u ,  v
>.  |  ( u  e.  c  /\  v  e.  c ) }
107, 8, 93sstr4i 3347 . . . . . 6  |-  S  C_  ( c  X.  c
)
11 ssdomg 7112 . . . . . 6  |-  ( ( c  X.  c )  e.  _V  ->  ( S  C_  ( c  X.  c )  ->  S  ~<_  ( c  X.  c
) ) )
125, 10, 11mp2 9 . . . . 5  |-  S  ~<_  ( c  X.  c )
134xpdom1 7166 . . . . . . . . 9  |-  ( c  ~<_  om  ->  ( c  X.  c )  ~<_  ( om 
X.  c ) )
14 omex 7554 . . . . . . . . . 10  |-  om  e.  _V
1514xpdom2 7162 . . . . . . . . 9  |-  ( c  ~<_  om  ->  ( om  X.  c )  ~<_  ( om 
X.  om ) )
16 domtr 7119 . . . . . . . . 9  |-  ( ( ( c  X.  c
)  ~<_  ( om  X.  c )  /\  ( om  X.  c )  ~<_  ( om  X.  om )
)  ->  ( c  X.  c )  ~<_  ( om 
X.  om ) )
1713, 15, 16syl2anc 643 . . . . . . . 8  |-  ( c  ~<_  om  ->  ( c  X.  c )  ~<_  ( om 
X.  om ) )
18 xpomen 7853 . . . . . . . 8  |-  ( om 
X.  om )  ~~  om
19 domentr 7125 . . . . . . . 8  |-  ( ( ( c  X.  c
)  ~<_  ( om  X.  om )  /\  ( om  X.  om )  ~~  om )  ->  ( c  X.  c )  ~<_  om )
2017, 18, 19sylancl 644 . . . . . . 7  |-  ( c  ~<_  om  ->  ( c  X.  c )  ~<_  om )
2120adantr 452 . . . . . 6  |-  ( ( c  ~<_  om  /\  ( topGen `
 c )  =  J )  ->  (
c  X.  c )  ~<_  om )
2221ad2antll 710 . . . . 5  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  ( c  X.  c )  ~<_  om )
23 domtr 7119 . . . . 5  |-  ( ( S  ~<_  ( c  X.  c )  /\  (
c  X.  c )  ~<_  om )  ->  S  ~<_  om )
2412, 22, 23sylancr 645 . . . 4  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  S  ~<_  om )
258relopabi 4959 . . . . . . . . 9  |-  Rel  S
26 simpr 448 . . . . . . . . 9  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  x  e.  S )  ->  x  e.  S )
27 1st2nd 6352 . . . . . . . . 9  |-  ( ( Rel  S  /\  x  e.  S )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2825, 26, 27sylancr 645 . . . . . . . 8  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  x  e.  S )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2928, 26eqeltrrd 2479 . . . . . . 7  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  x  e.  S )  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  S )
30 df-br 4173 . . . . . . . . 9  |-  ( ( 1st `  x ) S ( 2nd `  x
)  <->  <. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  S
)
31 fvex 5701 . . . . . . . . . 10  |-  ( 1st `  x )  e.  _V
32 fvex 5701 . . . . . . . . . 10  |-  ( 2nd `  x )  e.  _V
33 simpl 444 . . . . . . . . . . . 12  |-  ( ( u  =  ( 1st `  x )  /\  v  =  ( 2nd `  x
) )  ->  u  =  ( 1st `  x
) )
3433eleq1d 2470 . . . . . . . . . . 11  |-  ( ( u  =  ( 1st `  x )  /\  v  =  ( 2nd `  x
) )  ->  (
u  e.  c  <->  ( 1st `  x )  e.  c ) )
35 simpr 448 . . . . . . . . . . . 12  |-  ( ( u  =  ( 1st `  x )  /\  v  =  ( 2nd `  x
) )  ->  v  =  ( 2nd `  x
) )
3635eleq1d 2470 . . . . . . . . . . 11  |-  ( ( u  =  ( 1st `  x )  /\  v  =  ( 2nd `  x
) )  ->  (
v  e.  c  <->  ( 2nd `  x )  e.  c ) )
37 sseq1 3329 . . . . . . . . . . . . 13  |-  ( u  =  ( 1st `  x
)  ->  ( u  C_  w  <->  ( 1st `  x
)  C_  w )
)
38 sseq2 3330 . . . . . . . . . . . . 13  |-  ( v  =  ( 2nd `  x
)  ->  ( w  C_  v  <->  w  C_  ( 2nd `  x ) ) )
3937, 38bi2anan9 844 . . . . . . . . . . . 12  |-  ( ( u  =  ( 1st `  x )  /\  v  =  ( 2nd `  x
) )  ->  (
( u  C_  w  /\  w  C_  v )  <-> 
( ( 1st `  x
)  C_  w  /\  w  C_  ( 2nd `  x
) ) ) )
4039rexbidv 2687 . . . . . . . . . . 11  |-  ( ( u  =  ( 1st `  x )  /\  v  =  ( 2nd `  x
) )  ->  ( E. w  e.  B  ( u  C_  w  /\  w  C_  v )  <->  E. w  e.  B  ( ( 1st `  x )  C_  w  /\  w  C_  ( 2nd `  x ) ) ) )
4134, 36, 403anbi123d 1254 . . . . . . . . . 10  |-  ( ( u  =  ( 1st `  x )  /\  v  =  ( 2nd `  x
) )  ->  (
( u  e.  c  /\  v  e.  c  /\  E. w  e.  B  ( u  C_  w  /\  w  C_  v
) )  <->  ( ( 1st `  x )  e.  c  /\  ( 2nd `  x )  e.  c  /\  E. w  e.  B  ( ( 1st `  x )  C_  w  /\  w  C_  ( 2nd `  x ) ) ) ) )
4231, 32, 41, 8braba 4432 . . . . . . . . 9  |-  ( ( 1st `  x ) S ( 2nd `  x
)  <->  ( ( 1st `  x )  e.  c  /\  ( 2nd `  x
)  e.  c  /\  E. w  e.  B  ( ( 1st `  x
)  C_  w  /\  w  C_  ( 2nd `  x
) ) ) )
4330, 42bitr3i 243 . . . . . . . 8  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  S  <->  ( ( 1st `  x
)  e.  c  /\  ( 2nd `  x )  e.  c  /\  E. w  e.  B  (
( 1st `  x
)  C_  w  /\  w  C_  ( 2nd `  x
) ) ) )
4443simp3bi 974 . . . . . . 7  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  S  ->  E. w  e.  B  ( ( 1st `  x
)  C_  w  /\  w  C_  ( 2nd `  x
) ) )
4529, 44syl 16 . . . . . 6  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  x  e.  S )  ->  E. w  e.  B  ( ( 1st `  x )  C_  w  /\  w  C_  ( 2nd `  x ) ) )
46 fvi 5742 . . . . . . . 8  |-  ( B  e.  TopBases  ->  (  _I  `  B )  =  B )
4746ad3antrrr 711 . . . . . . 7  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  x  e.  S )  ->  (  _I  `  B )  =  B )
4847rexeqdv 2871 . . . . . 6  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  x  e.  S )  ->  ( E. w  e.  (  _I  `  B ) ( ( 1st `  x
)  C_  w  /\  w  C_  ( 2nd `  x
) )  <->  E. w  e.  B  ( ( 1st `  x )  C_  w  /\  w  C_  ( 2nd `  x ) ) ) )
4945, 48mpbird 224 . . . . 5  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  x  e.  S )  ->  E. w  e.  (  _I  `  B
) ( ( 1st `  x )  C_  w  /\  w  C_  ( 2nd `  x ) ) )
5049ralrimiva 2749 . . . 4  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  A. x  e.  S  E. w  e.  (  _I  `  B ) ( ( 1st `  x
)  C_  w  /\  w  C_  ( 2nd `  x
) ) )
51 fvex 5701 . . . . 5  |-  (  _I 
`  B )  e. 
_V
52 sseq2 3330 . . . . . 6  |-  ( w  =  ( f `  x )  ->  (
( 1st `  x
)  C_  w  <->  ( 1st `  x )  C_  (
f `  x )
) )
53 sseq1 3329 . . . . . 6  |-  ( w  =  ( f `  x )  ->  (
w  C_  ( 2nd `  x )  <->  ( f `  x )  C_  ( 2nd `  x ) ) )
5452, 53anbi12d 692 . . . . 5  |-  ( w  =  ( f `  x )  ->  (
( ( 1st `  x
)  C_  w  /\  w  C_  ( 2nd `  x
) )  <->  ( ( 1st `  x )  C_  ( f `  x
)  /\  ( f `  x )  C_  ( 2nd `  x ) ) ) )
5551, 54axcc4dom 8277 . . . 4  |-  ( ( S  ~<_  om  /\  A. x  e.  S  E. w  e.  (  _I  `  B
) ( ( 1st `  x )  C_  w  /\  w  C_  ( 2nd `  x ) ) )  ->  E. f ( f : S --> (  _I 
`  B )  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )
5624, 50, 55syl2anc 643 . . 3  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  E. f ( f : S --> (  _I 
`  B )  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )
5746ad2antrr 707 . . . . . . 7  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  (  _I  `  B )  =  B )
58 feq3 5537 . . . . . . 7  |-  ( (  _I  `  B )  =  B  ->  (
f : S --> (  _I 
`  B )  <->  f : S
--> B ) )
5957, 58syl 16 . . . . . 6  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  ( f : S --> (  _I  `  B )  <->  f : S
--> B ) )
6059anbi1d 686 . . . . 5  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  ( ( f : S --> (  _I 
`  B )  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  <->  ( f : S --> B  /\  A. x  e.  S  (
( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) ) )
61 2ndctop 17463 . . . . . . . . . . . 12  |-  ( J  e.  2ndc  ->  J  e. 
Top )
6261adantl 453 . . . . . . . . . . 11  |-  ( ( B  e.  TopBases  /\  J  e.  2ndc )  ->  J  e.  Top )
6362ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  J  e.  Top )
64 frn 5556 . . . . . . . . . . . 12  |-  ( f : S --> B  ->  ran  f  C_  B )
6564ad2antrl 709 . . . . . . . . . . 11  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  ran  f  C_  B )
66 bastg 16986 . . . . . . . . . . . . 13  |-  ( B  e.  TopBases  ->  B  C_  ( topGen `
 B ) )
6766ad3antrrr 711 . . . . . . . . . . . 12  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  B  C_  ( topGen `
 B ) )
68 2ndcctbss.2 . . . . . . . . . . . 12  |-  J  =  ( topGen `  B )
6967, 68syl6sseqr 3355 . . . . . . . . . . 11  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  B  C_  J
)
7065, 69sstrd 3318 . . . . . . . . . 10  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  ran  f  C_  J )
71 simprrl 741 . . . . . . . . . . . . . . 15  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  o  e.  J
)
72 simprr 734 . . . . . . . . . . . . . . . 16  |-  ( ( c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) )  -> 
( topGen `  c )  =  J )
7372ad2antlr 708 . . . . . . . . . . . . . . 15  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  ( topGen `  c
)  =  J )
7471, 73eleqtrrd 2481 . . . . . . . . . . . . . 14  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  o  e.  (
topGen `  c ) )
75 simprrr 742 . . . . . . . . . . . . . 14  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  t  e.  o )
76 tg2 16985 . . . . . . . . . . . . . 14  |-  ( ( o  e.  ( topGen `  c )  /\  t  e.  o )  ->  E. d  e.  c  ( t  e.  d  /\  d  C_  o ) )
7774, 75, 76syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  E. d  e.  c  ( t  e.  d  /\  d  C_  o
) )
78 bastg 16986 . . . . . . . . . . . . . . . . . . 19  |-  ( c  e.  TopBases  ->  c  C_  ( topGen `
 c ) )
7978ad2antrl 709 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  c  C_  ( topGen `
 c ) )
8079ad2antrr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  c  C_  ( topGen `
 c ) )
8168eqeq2i 2414 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
topGen `  c )  =  J  <->  ( topGen `  c
)  =  ( topGen `  B ) )
8281biimpi 187 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
topGen `  c )  =  J  ->  ( topGen `  c )  =  (
topGen `  B ) )
8382adantl 453 . . . . . . . . . . . . . . . . . . 19  |-  ( ( c  ~<_  om  /\  ( topGen `
 c )  =  J )  ->  ( topGen `
 c )  =  ( topGen `  B )
)
8483ad2antll 710 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  ( topGen `  c
)  =  ( topGen `  B ) )
8584ad2antrr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  ( topGen `  c
)  =  ( topGen `  B ) )
8680, 85sseqtrd 3344 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  c  C_  ( topGen `
 B ) )
87 simprl 733 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  d  e.  c )
8886, 87sseldd 3309 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  d  e.  (
topGen `  B ) )
89 simprrl 741 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  t  e.  d )
90 tg2 16985 . . . . . . . . . . . . . . 15  |-  ( ( d  e.  ( topGen `  B )  /\  t  e.  d )  ->  E. m  e.  B  ( t  e.  m  /\  m  C_  d ) )
9188, 89, 90syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  E. m  e.  B  ( t  e.  m  /\  m  C_  d ) )
9266ad3antrrr 711 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  B  C_  ( topGen `
 B ) )
9392ad2antrr 707 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  B  C_  ( topGen `
 B ) )
9473ad2antrr 707 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  ( topGen `  c
)  =  J )
9594, 68syl6req 2453 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  ( topGen `  B
)  =  ( topGen `  c ) )
9693, 95sseqtrd 3344 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  B  C_  ( topGen `
 c ) )
97 simprl 733 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  m  e.  B
)
9896, 97sseldd 3309 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  m  e.  (
topGen `  c ) )
99 simprrl 741 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  t  e.  m
)
100 tg2 16985 . . . . . . . . . . . . . . . 16  |-  ( ( m  e.  ( topGen `  c )  /\  t  e.  m )  ->  E. n  e.  c  ( t  e.  n  /\  n  C_  m ) )
10198, 99, 100syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  E. n  e.  c  ( t  e.  n  /\  n  C_  m ) )
102 ffn 5550 . . . . . . . . . . . . . . . . . . . 20  |-  ( f : S --> B  -> 
f  Fn  S )
103102ad2antrr 707 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) )  -> 
f  Fn  S )
104103ad2antlr 708 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  f  Fn  S
)
105104ad2antrr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  f  Fn  S
)
106 simprl 733 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  n  e.  c )
10787ad2antrr 707 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  d  e.  c )
108 simplrl 737 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  m  e.  B
)
109 simprrr 742 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  n  C_  m
)
110 simprr 734 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) )  ->  m  C_  d
)
111110ad2antlr 708 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  m  C_  d
)
112 sseq2 3330 . . . . . . . . . . . . . . . . . . . . 21  |-  ( w  =  m  ->  (
n  C_  w  <->  n  C_  m
) )
113 sseq1 3329 . . . . . . . . . . . . . . . . . . . . 21  |-  ( w  =  m  ->  (
w  C_  d  <->  m  C_  d
) )
114112, 113anbi12d 692 . . . . . . . . . . . . . . . . . . . 20  |-  ( w  =  m  ->  (
( n  C_  w  /\  w  C_  d )  <-> 
( n  C_  m  /\  m  C_  d ) ) )
115114rspcev 3012 . . . . . . . . . . . . . . . . . . 19  |-  ( ( m  e.  B  /\  ( n  C_  m  /\  m  C_  d ) )  ->  E. w  e.  B  ( n  C_  w  /\  w  C_  d ) )
116108, 109, 111, 115syl12anc 1182 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  E. w  e.  B  ( n  C_  w  /\  w  C_  d ) )
117 df-br 4173 . . . . . . . . . . . . . . . . . . 19  |-  ( n S d  <->  <. n ,  d >.  e.  S
)
118 vex 2919 . . . . . . . . . . . . . . . . . . . 20  |-  n  e. 
_V
119 vex 2919 . . . . . . . . . . . . . . . . . . . 20  |-  d  e. 
_V
120 simpl 444 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( u  =  n  /\  v  =  d )  ->  u  =  n )
121120eleq1d 2470 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( u  =  n  /\  v  =  d )  ->  ( u  e.  c  <-> 
n  e.  c ) )
122 simpr 448 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( u  =  n  /\  v  =  d )  ->  v  =  d )
123122eleq1d 2470 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( u  =  n  /\  v  =  d )  ->  ( v  e.  c  <-> 
d  e.  c ) )
124 sseq1 3329 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( u  =  n  ->  (
u  C_  w  <->  n  C_  w
) )
125 sseq2 3330 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( v  =  d  ->  (
w  C_  v  <->  w  C_  d
) )
126124, 125bi2anan9 844 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( u  =  n  /\  v  =  d )  ->  ( ( u  C_  w  /\  w  C_  v
)  <->  ( n  C_  w  /\  w  C_  d
) ) )
127126rexbidv 2687 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( u  =  n  /\  v  =  d )  ->  ( E. w  e.  B  ( u  C_  w  /\  w  C_  v
)  <->  E. w  e.  B  ( n  C_  w  /\  w  C_  d ) ) )
128121, 123, 1273anbi123d 1254 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( u  =  n  /\  v  =  d )  ->  ( ( u  e.  c  /\  v  e.  c  /\  E. w  e.  B  ( u  C_  w  /\  w  C_  v ) )  <->  ( n  e.  c  /\  d  e.  c  /\  E. w  e.  B  ( n  C_  w  /\  w  C_  d ) ) ) )
129118, 119, 128, 8braba 4432 . . . . . . . . . . . . . . . . . . 19  |-  ( n S d  <->  ( n  e.  c  /\  d  e.  c  /\  E. w  e.  B  ( n  C_  w  /\  w  C_  d ) ) )
130117, 129bitr3i 243 . . . . . . . . . . . . . . . . . 18  |-  ( <.
n ,  d >.  e.  S  <->  ( n  e.  c  /\  d  e.  c  /\  E. w  e.  B  ( n  C_  w  /\  w  C_  d ) ) )
131106, 107, 116, 130syl3anbrc 1138 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  <. n ,  d
>.  e.  S )
132 fnfvelrn 5826 . . . . . . . . . . . . . . . . 17  |-  ( ( f  Fn  S  /\  <.
n ,  d >.  e.  S )  ->  (
f `  <. n ,  d >. )  e.  ran  f )
133105, 131, 132syl2anc 643 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  ( f `  <. n ,  d >.
)  e.  ran  f
)
134 simprl 733 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  n  e.  c )
135 simplll 735 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  d  e.  c )
136 simplrl 737 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  m  e.  B )
137 simprrr 742 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  n  C_  m )
138110ad2antlr 708 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  m  C_  d )
139136, 137, 138, 115syl12anc 1182 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  E. w  e.  B  ( n  C_  w  /\  w  C_  d ) )
140134, 135, 139, 130syl3anbrc 1138 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  <. n ,  d >.  e.  S
)
141 fveq2 5687 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( x  =  <. n ,  d
>.  ->  ( 1st `  x
)  =  ( 1st `  <. n ,  d
>. ) )
142 fveq2 5687 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( x  =  <. n ,  d
>.  ->  ( f `  x )  =  ( f `  <. n ,  d >. )
)
143141, 142sseq12d 3337 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( x  =  <. n ,  d
>.  ->  ( ( 1st `  x )  C_  (
f `  x )  <->  ( 1st `  <. n ,  d >. )  C_  ( f `  <. n ,  d >. )
) )
144 fveq2 5687 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( x  =  <. n ,  d
>.  ->  ( 2nd `  x
)  =  ( 2nd `  <. n ,  d
>. ) )
145142, 144sseq12d 3337 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( x  =  <. n ,  d
>.  ->  ( ( f `
 x )  C_  ( 2nd `  x )  <-> 
( f `  <. n ,  d >. )  C_  ( 2nd `  <. n ,  d >. )
) )
146143, 145anbi12d 692 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  <. n ,  d
>.  ->  ( ( ( 1st `  x ) 
C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) )  <->  ( ( 1st `  <. n ,  d
>. )  C_  ( f `
 <. n ,  d
>. )  /\  (
f `  <. n ,  d >. )  C_  ( 2nd `  <. n ,  d
>. ) ) ) )
147146rspcv 3008 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
n ,  d >.  e.  S  ->  ( A. x  e.  S  (
( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) )  ->  (
( 1st `  <. n ,  d >. )  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  ( 2nd `  <. n ,  d >. )
) ) )
148140, 147syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  ( A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) )  ->  (
( 1st `  <. n ,  d >. )  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  ( 2nd `  <. n ,  d >. )
) ) )
149118, 119op1st 6314 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 1st `  <. n ,  d
>. )  =  n
150149sseq1i 3332 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( 1st `  <. n ,  d >. )  C_  ( f `  <. n ,  d >. )  <->  n 
C_  ( f `  <. n ,  d >.
) )
151118, 119op2nd 6315 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 2nd `  <. n ,  d
>. )  =  d
152151sseq2i 3333 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( f `  <. n ,  d >. )  C_  ( 2nd `  <. n ,  d >. )  <->  ( f `  <. n ,  d >. )  C_  d )
153150, 152anbi12i 679 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( 1st `  <. n ,  d >. )  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  ( 2nd `  <. n ,  d >. )
)  <->  ( n  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  d ) )
154 simprl 733 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  /\  ( n  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  d ) )  ->  n  C_  ( f `  <. n ,  d >.
) )
155 simprl 733 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) )  ->  t  e.  n )
156155ad2antlr 708 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  /\  ( n  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  d ) )  -> 
t  e.  n )
157154, 156sseldd 3309 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  /\  ( n  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  d ) )  -> 
t  e.  ( f `
 <. n ,  d
>. ) )
158 simprr 734 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  /\  ( n  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  d ) )  -> 
( f `  <. n ,  d >. )  C_  d )
159 simplrr 738 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  d  C_  o )
160159ad2antrr 707 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  /\  ( n  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  d ) )  -> 
d  C_  o )
161158, 160sstrd 3318 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  /\  ( n  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  d ) )  -> 
( f `  <. n ,  d >. )  C_  o )
162157, 161jca 519 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  /\  ( n  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  d ) )  -> 
( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) )
163162ex 424 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  (
( n  C_  (
f `  <. n ,  d >. )  /\  (
f `  <. n ,  d >. )  C_  d
)  ->  ( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) ) )
164153, 163syl5bi 209 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  (
( ( 1st `  <. n ,  d >. )  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  ( 2nd `  <. n ,  d >. )
)  ->  ( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) ) )
165148, 164syld 42 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  ( A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) )  ->  (
t  e.  ( f `
 <. n ,  d
>. )  /\  (
f `  <. n ,  d >. )  C_  o
) ) )
166165com12 29 . . . . . . . . . . . . . . . . . . . 20  |-  ( A. x  e.  S  (
( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) )  ->  (
( ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  ( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) ) )
167166exp4c 592 . . . . . . . . . . . . . . . . . . 19  |-  ( A. x  e.  S  (
( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) )  ->  (
( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  -> 
( ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) )  -> 
( ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) )  -> 
( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) ) ) ) )
168167ad2antlr 708 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) )  -> 
( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  -> 
( ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) )  -> 
( ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) )  -> 
( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) ) ) ) )
169168adantl 453 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  ->  ( ( m  e.  B  /\  (
t  e.  m  /\  m  C_  d ) )  ->  ( ( n  e.  c  /\  (
t  e.  n  /\  n  C_  m ) )  ->  ( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) ) ) ) )
170169imp41 577 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  ( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) )
171 eleq2 2465 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( f `  <. n ,  d >.
)  ->  ( t  e.  b  <->  t  e.  ( f `  <. n ,  d >. )
) )
172 sseq1 3329 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( f `  <. n ,  d >.
)  ->  ( b  C_  o  <->  ( f `  <. n ,  d >.
)  C_  o )
)
173171, 172anbi12d 692 . . . . . . . . . . . . . . . . 17  |-  ( b  =  ( f `  <. n ,  d >.
)  ->  ( (
t  e.  b  /\  b  C_  o )  <->  ( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) ) )
174173rspcev 3012 . . . . . . . . . . . . . . . 16  |-  ( ( ( f `  <. n ,  d >. )  e.  ran  f  /\  (
t  e.  ( f `
 <. n ,  d
>. )  /\  (
f `  <. n ,  d >. )  C_  o
) )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) )
175133, 170, 174syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) )
176101, 175rexlimddv 2794 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) )
17791, 176rexlimddv 2794 . . . . . . . . . . . . 13  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) )
17877, 177rexlimddv 2794 . . . . . . . . . . . 12  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) )
179178expr 599 . . . . . . . . . . 11  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  ( ( o  e.  J  /\  t  e.  o )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) ) )
180179ralrimivv 2757 . . . . . . . . . 10  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  A. o  e.  J  A. t  e.  o  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) )
181 basgen2 17009 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ran  f  C_  J  /\  A. o  e.  J  A. t  e.  o  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) )  ->  ( topGen `  ran  f )  =  J )
18263, 70, 180, 181syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  ( topGen `  ran  f )  =  J )
183182, 63eqeltrd 2478 . . . . . . . 8  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  ( topGen `  ran  f )  e.  Top )
184 tgclb 16990 . . . . . . . 8  |-  ( ran  f  e.  TopBases  <->  ( topGen ` 
ran  f )  e. 
Top )
185183, 184sylibr 204 . . . . . . 7  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  ran  f  e.  TopBases )
186 omelon 7557 . . . . . . . . . 10  |-  om  e.  On
18724adantr 452 . . . . . . . . . 10  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  S  ~<_  om )
188 ondomen 7874 . . . . . . . . . 10  |-  ( ( om  e.  On  /\  S  ~<_  om )  ->  S  e.  dom  card )
189186, 187, 188sylancr 645 . . . . . . . . 9  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  S  e.  dom  card )
190102ad2antrl 709 . . . . . . . . . 10  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  f  Fn  S
)
191 dffn4 5618 . . . . . . . . . 10  |-  ( f  Fn  S  <->  f : S -onto-> ran  f )
192190, 191sylib 189 . . . . . . . . 9  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  f : S -onto-> ran  f )
193 fodomnum 7894 . . . . . . . . 9  |-  ( S  e.  dom  card  ->  ( f : S -onto-> ran  f  ->  ran  f  ~<_  S ) )
194189, 192, 193sylc 58 . . . . . . . 8  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  ran  f  ~<_  S )
195 domtr 7119 . . . . . . . 8  |-  ( ( ran  f  ~<_  S  /\  S  ~<_  om )  ->  ran  f  ~<_  om )
196194, 187, 195syl2anc 643 . . . . . . 7  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  ran  f  ~<_  om )
197182eqcomd 2409 . . . . . . 7  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  J  =  (
topGen `  ran  f ) )
198 breq1 4175 . . . . . . . . 9  |-  ( b  =  ran  f  -> 
( b  ~<_  om  <->  ran  f  ~<_  om ) )
199 sseq1 3329 . . . . . . . . 9  |-  ( b  =  ran  f  -> 
( b  C_  B  <->  ran  f  C_  B )
)
200 fveq2 5687 . . . . . . . . . 10  |-  ( b  =  ran  f  -> 
( topGen `  b )  =  ( topGen `  ran  f ) )
201200eqeq2d 2415 . . . . . . . . 9  |-  ( b  =  ran  f  -> 
( J  =  (
topGen `  b )  <->  J  =  ( topGen `  ran  f ) ) )
202198, 199, 2013anbi123d 1254 . . . . . . . 8  |-  ( b  =  ran  f  -> 
( ( b  ~<_  om 
/\  b  C_  B  /\  J  =  ( topGen `
 b ) )  <-> 
( ran  f  ~<_  om  /\  ran  f  C_  B  /\  J  =  ( topGen ` 
ran  f ) ) ) )
203202rspcev 3012 . . . . . . 7  |-  ( ( ran  f  e.  TopBases  /\  ( ran  f  ~<_  om  /\  ran  f  C_  B  /\  J  =  ( topGen ` 
ran  f ) ) )  ->  E. b  e. 
TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  ( topGen `  b
) ) )
204185, 196, 65, 197, 203syl13anc 1186 . . . . . 6  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  E. b  e.  TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  ( topGen `  b )
) )
205204ex 424 . . . . 5  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  ->  E. b  e.  TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  (
topGen `  b ) ) ) )
20660, 205sylbid 207 . . . 4  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  ( ( f : S --> (  _I 
`  B )  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  ->  E. b  e.  TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  (
topGen `  b ) ) ) )
207206exlimdv 1643 . . 3  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  ( E. f
( f : S --> (  _I  `  B )  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  ->  E. b  e.  TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  ( topGen `  b )
) ) )
20856, 207mpd 15 . 2  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  E. b  e.  TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  ( topGen `  b )
) )
2093, 208rexlimddv 2794 1  |-  ( ( B  e.  TopBases  /\  J  e.  2ndc )  ->  E. b  e. 
TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  ( topGen `  b
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916    C_ wss 3280   <.cop 3777   U.cuni 3975   class class class wbr 4172   {copab 4225    _I cid 4453   Oncon0 4541   omcom 4804    X. cxp 4835   dom cdm 4837   ran crn 4838   Rel wrel 4842    Fn wfn 5408   -->wf 5409   -onto->wfo 5411   ` cfv 5413   1stc1st 6306   2ndc2nd 6307    ~~ cen 7065    ~<_ cdom 7066   cardccrd 7778   topGenctg 13620   Topctop 16913   TopBasesctb 16917   2ndcc2ndc 17454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cc 8271
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-card 7782  df-acn 7785  df-topgen 13622  df-top 16918  df-bases 16920  df-2ndc 17456
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