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Theorem utop3cls 19831
Description: Relation between a topological closure and a symmetric entourage in an uniform space. Second part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Jan-2018.)
Hypothesis
Ref Expression
utoptop.1  |-  J  =  (unifTop `  U )
Assertion
Ref Expression
utop3cls  |-  ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X
) )  /\  ( V  e.  U  /\  `' V  =  V
) )  ->  (
( cls `  ( J  tX  J ) ) `
 M )  C_  ( V  o.  ( M  o.  V )
) )

Proof of Theorem utop3cls
Dummy variables  r 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 4952 . . . . 5  |-  Rel  ( X  X.  X )
2 utoptop.1 . . . . . . . . . . 11  |-  J  =  (unifTop `  U )
3 utoptop 19814 . . . . . . . . . . 11  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  e.  Top )
42, 3syl5eqel 2527 . . . . . . . . . 10  |-  ( U  e.  (UnifOn `  X
)  ->  J  e.  Top )
5 txtop 19147 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  J  e.  Top )  ->  ( J  tX  J
)  e.  Top )
64, 4, 5syl2anc 661 . . . . . . . . 9  |-  ( U  e.  (UnifOn `  X
)  ->  ( J  tX  J )  e.  Top )
76ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  -> 
( J  tX  J
)  e.  Top )
8 simpllr 758 . . . . . . . . 9  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  ->  M  C_  ( X  X.  X ) )
9 utoptopon 19816 . . . . . . . . . . . . . 14  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  e.  (TopOn `  X ) )
102, 9syl5eqel 2527 . . . . . . . . . . . . 13  |-  ( U  e.  (UnifOn `  X
)  ->  J  e.  (TopOn `  X ) )
11 toponuni 18537 . . . . . . . . . . . . 13  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1210, 11syl 16 . . . . . . . . . . . 12  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. J )
1312, 12xpeq12d 4870 . . . . . . . . . . 11  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  =  ( U. J  X.  U. J ) )
14 eqid 2443 . . . . . . . . . . . . 13  |-  U. J  =  U. J
1514, 14txuni 19170 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  J  e.  Top )  ->  ( U. J  X.  U. J )  =  U. ( J  tX  J ) )
164, 4, 15syl2anc 661 . . . . . . . . . . 11  |-  ( U  e.  (UnifOn `  X
)  ->  ( U. J  X.  U. J )  =  U. ( J 
tX  J ) )
1713, 16eqtrd 2475 . . . . . . . . . 10  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  =  U. ( J  tX  J ) )
1817ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  -> 
( X  X.  X
)  =  U. ( J  tX  J ) )
198, 18sseqtrd 3397 . . . . . . . 8  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  ->  M  C_  U. ( J 
tX  J ) )
20 eqid 2443 . . . . . . . . 9  |-  U. ( J  tX  J )  = 
U. ( J  tX  J )
2120clsss3 18668 . . . . . . . 8  |-  ( ( ( J  tX  J
)  e.  Top  /\  M  C_  U. ( J 
tX  J ) )  ->  ( ( cls `  ( J  tX  J
) ) `  M
)  C_  U. ( J  tX  J ) )
227, 19, 21syl2anc 661 . . . . . . 7  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  -> 
( ( cls `  ( J  tX  J ) ) `
 M )  C_  U. ( J  tX  J
) )
2322, 18sseqtr4d 3398 . . . . . 6  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  -> 
( ( cls `  ( J  tX  J ) ) `
 M )  C_  ( X  X.  X
) )
24 simpr 461 . . . . . 6  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  -> 
z  e.  ( ( cls `  ( J 
tX  J ) ) `
 M ) )
2523, 24sseldd 3362 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  -> 
z  e.  ( X  X.  X ) )
26 1st2nd 6625 . . . . 5  |-  ( ( Rel  ( X  X.  X )  /\  z  e.  ( X  X.  X
) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
271, 25, 26sylancr 663 . . . 4  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  -> 
z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
28 simp-4l 765 . . . . . . . . . 10  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  U  e.  (UnifOn `  X ) )
29 simpr1l 1045 . . . . . . . . . . 11  |-  ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X
) )  /\  (
( V  e.  U  /\  `' V  =  V
)  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `
 M )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) ) )  ->  V  e.  U )
30293anassrs 1209 . . . . . . . . . 10  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  V  e.  U
)
31 ustrel 19791 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  Rel  V )
3228, 30, 31syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  Rel  V )
33 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  r  e.  ( ( ( V " { ( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )
34 elin 3544 . . . . . . . . . . . 12  |-  ( r  e.  ( ( ( V " { ( 1st `  z ) } )  X.  ( V " { ( 2nd `  z ) } ) )  i^i  M )  <-> 
( r  e.  ( ( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  /\  r  e.  M
) )
3533, 34sylib 196 . . . . . . . . . . 11  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  ( r  e.  ( ( V " { ( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  /\  r  e.  M
) )
3635simpld 459 . . . . . . . . . 10  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  r  e.  ( ( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) ) )
37 xp1st 6611 . . . . . . . . . 10  |-  ( r  e.  ( ( V
" { ( 1st `  z ) } )  X.  ( V " { ( 2nd `  z
) } ) )  ->  ( 1st `  r
)  e.  ( V
" { ( 1st `  z ) } ) )
3836, 37syl 16 . . . . . . . . 9  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  ( 1st `  r
)  e.  ( V
" { ( 1st `  z ) } ) )
39 elrelimasn 5198 . . . . . . . . . 10  |-  ( Rel 
V  ->  ( ( 1st `  r )  e.  ( V " {
( 1st `  z
) } )  <->  ( 1st `  z ) V ( 1st `  r ) ) )
4039biimpa 484 . . . . . . . . 9  |-  ( ( Rel  V  /\  ( 1st `  r )  e.  ( V " {
( 1st `  z
) } ) )  ->  ( 1st `  z
) V ( 1st `  r ) )
4132, 38, 40syl2anc 661 . . . . . . . 8  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  ( 1st `  z
) V ( 1st `  r ) )
42 simp-4r 766 . . . . . . . . . . 11  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  M  C_  ( X  X.  X ) )
43 xpss 4951 . . . . . . . . . . 11  |-  ( X  X.  X )  C_  ( _V  X.  _V )
4442, 43syl6ss 3373 . . . . . . . . . 10  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  M  C_  ( _V  X.  _V ) )
45 df-rel 4852 . . . . . . . . . 10  |-  ( Rel 
M  <->  M  C_  ( _V 
X.  _V ) )
4644, 45sylibr 212 . . . . . . . . 9  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  Rel  M )
4735simprd 463 . . . . . . . . 9  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  r  e.  M
)
48 1st2ndbr 6628 . . . . . . . . 9  |-  ( ( Rel  M  /\  r  e.  M )  ->  ( 1st `  r ) M ( 2nd `  r
) )
4946, 47, 48syl2anc 661 . . . . . . . 8  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  ( 1st `  r
) M ( 2nd `  r ) )
50 xp2nd 6612 . . . . . . . . . . 11  |-  ( r  e.  ( ( V
" { ( 1st `  z ) } )  X.  ( V " { ( 2nd `  z
) } ) )  ->  ( 2nd `  r
)  e.  ( V
" { ( 2nd `  z ) } ) )
5136, 50syl 16 . . . . . . . . . 10  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  ( 2nd `  r
)  e.  ( V
" { ( 2nd `  z ) } ) )
52 elrelimasn 5198 . . . . . . . . . . 11  |-  ( Rel 
V  ->  ( ( 2nd `  r )  e.  ( V " {
( 2nd `  z
) } )  <->  ( 2nd `  z ) V ( 2nd `  r ) ) )
5352biimpa 484 . . . . . . . . . 10  |-  ( ( Rel  V  /\  ( 2nd `  r )  e.  ( V " {
( 2nd `  z
) } ) )  ->  ( 2nd `  z
) V ( 2nd `  r ) )
5432, 51, 53syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  ( 2nd `  z
) V ( 2nd `  r ) )
55 simpr1r 1046 . . . . . . . . . . 11  |-  ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X
) )  /\  (
( V  e.  U  /\  `' V  =  V
)  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `
 M )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) ) )  ->  `' V  =  V )
56553anassrs 1209 . . . . . . . . . 10  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  `' V  =  V )
57 fvex 5706 . . . . . . . . . . . 12  |-  ( 2nd `  r )  e.  _V
58 fvex 5706 . . . . . . . . . . . 12  |-  ( 2nd `  z )  e.  _V
5957, 58brcnv 5027 . . . . . . . . . . 11  |-  ( ( 2nd `  r ) `' V ( 2nd `  z
)  <->  ( 2nd `  z
) V ( 2nd `  r ) )
60 breq 4299 . . . . . . . . . . 11  |-  ( `' V  =  V  -> 
( ( 2nd `  r
) `' V ( 2nd `  z )  <-> 
( 2nd `  r
) V ( 2nd `  z ) ) )
6159, 60syl5rbbr 260 . . . . . . . . . 10  |-  ( `' V  =  V  -> 
( ( 2nd `  r
) V ( 2nd `  z )  <->  ( 2nd `  z ) V ( 2nd `  r ) ) )
6256, 61syl 16 . . . . . . . . 9  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  ( ( 2nd `  r ) V ( 2nd `  z )  <-> 
( 2nd `  z
) V ( 2nd `  r ) ) )
6354, 62mpbird 232 . . . . . . . 8  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  ( 2nd `  r
) V ( 2nd `  z ) )
64 fvex 5706 . . . . . . . . . 10  |-  ( 1st `  z )  e.  _V
65 fvex 5706 . . . . . . . . . 10  |-  ( 1st `  r )  e.  _V
66 brcogw 5013 . . . . . . . . . . 11  |-  ( ( ( ( 1st `  z
)  e.  _V  /\  ( 2nd `  r )  e.  _V  /\  ( 1st `  r )  e. 
_V )  /\  (
( 1st `  z
) V ( 1st `  r )  /\  ( 1st `  r ) M ( 2nd `  r
) ) )  -> 
( 1st `  z
) ( M  o.  V ) ( 2nd `  r ) )
6766ex 434 . . . . . . . . . 10  |-  ( ( ( 1st `  z
)  e.  _V  /\  ( 2nd `  r )  e.  _V  /\  ( 1st `  r )  e. 
_V )  ->  (
( ( 1st `  z
) V ( 1st `  r )  /\  ( 1st `  r ) M ( 2nd `  r
) )  ->  ( 1st `  z ) ( M  o.  V ) ( 2nd `  r
) ) )
6864, 57, 65, 67mp3an 1314 . . . . . . . . 9  |-  ( ( ( 1st `  z
) V ( 1st `  r )  /\  ( 1st `  r ) M ( 2nd `  r
) )  ->  ( 1st `  z ) ( M  o.  V ) ( 2nd `  r
) )
69 brcogw 5013 . . . . . . . . . . 11  |-  ( ( ( ( 1st `  z
)  e.  _V  /\  ( 2nd `  z )  e.  _V  /\  ( 2nd `  r )  e. 
_V )  /\  (
( 1st `  z
) ( M  o.  V ) ( 2nd `  r )  /\  ( 2nd `  r ) V ( 2nd `  z
) ) )  -> 
( 1st `  z
) ( V  o.  ( M  o.  V
) ) ( 2nd `  z ) )
7069ex 434 . . . . . . . . . 10  |-  ( ( ( 1st `  z
)  e.  _V  /\  ( 2nd `  z )  e.  _V  /\  ( 2nd `  r )  e. 
_V )  ->  (
( ( 1st `  z
) ( M  o.  V ) ( 2nd `  r )  /\  ( 2nd `  r ) V ( 2nd `  z
) )  ->  ( 1st `  z ) ( V  o.  ( M  o.  V ) ) ( 2nd `  z
) ) )
7164, 58, 57, 70mp3an 1314 . . . . . . . . 9  |-  ( ( ( 1st `  z
) ( M  o.  V ) ( 2nd `  r )  /\  ( 2nd `  r ) V ( 2nd `  z
) )  ->  ( 1st `  z ) ( V  o.  ( M  o.  V ) ) ( 2nd `  z
) )
7268, 71sylan 471 . . . . . . . 8  |-  ( ( ( ( 1st `  z
) V ( 1st `  r )  /\  ( 1st `  r ) M ( 2nd `  r
) )  /\  ( 2nd `  r ) V ( 2nd `  z
) )  ->  ( 1st `  z ) ( V  o.  ( M  o.  V ) ) ( 2nd `  z
) )
7341, 49, 63, 72syl21anc 1217 . . . . . . 7  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  /\  r  e.  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) )  ->  ( 1st `  z
) ( V  o.  ( M  o.  V
) ) ( 2nd `  z ) )
7473ralrimiva 2804 . . . . . 6  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  ->  A. r  e.  (
( ( V " { ( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) ( 1st `  z ) ( V  o.  ( M  o.  V )
) ( 2nd `  z
) )
75 simplll 757 . . . . . . . . 9  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  ->  U  e.  (UnifOn `  X
) )
76 simplrl 759 . . . . . . . . 9  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  ->  V  e.  U )
7743ad2ant1 1009 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  z  e.  ( X  X.  X
) )  ->  J  e.  Top )
78 xp1st 6611 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  X )  ->  ( 1st `  z )  e.  X )
792utopsnnei 19829 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  ( 1st `  z )  e.  X )  ->  ( V " { ( 1st `  z ) } )  e.  ( ( nei `  J ) `  {
( 1st `  z
) } ) )
8078, 79syl3an3 1253 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  z  e.  ( X  X.  X
) )  ->  ( V " { ( 1st `  z ) } )  e.  ( ( nei `  J ) `  {
( 1st `  z
) } ) )
81 xp2nd 6612 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  X )  ->  ( 2nd `  z )  e.  X )
822utopsnnei 19829 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  ( 2nd `  z )  e.  X )  ->  ( V " { ( 2nd `  z ) } )  e.  ( ( nei `  J ) `  {
( 2nd `  z
) } ) )
8381, 82syl3an3 1253 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  z  e.  ( X  X.  X
) )  ->  ( V " { ( 2nd `  z ) } )  e.  ( ( nei `  J ) `  {
( 2nd `  z
) } ) )
8414, 14neitx 19185 . . . . . . . . . . 11  |-  ( ( ( J  e.  Top  /\  J  e.  Top )  /\  ( ( V " { ( 1st `  z
) } )  e.  ( ( nei `  J
) `  { ( 1st `  z ) } )  /\  ( V
" { ( 2nd `  z ) } )  e.  ( ( nei `  J ) `  {
( 2nd `  z
) } ) ) )  ->  ( ( V " { ( 1st `  z ) } )  X.  ( V " { ( 2nd `  z
) } ) )  e.  ( ( nei `  ( J  tX  J
) ) `  ( { ( 1st `  z
) }  X.  {
( 2nd `  z
) } ) ) )
8577, 77, 80, 83, 84syl22anc 1219 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  z  e.  ( X  X.  X
) )  ->  (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  e.  ( ( nei `  ( J  tX  J
) ) `  ( { ( 1st `  z
) }  X.  {
( 2nd `  z
) } ) ) )
86 1st2nd2 6618 . . . . . . . . . . . . . 14  |-  ( z  e.  ( X  X.  X )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
8786sneqd 3894 . . . . . . . . . . . . 13  |-  ( z  e.  ( X  X.  X )  ->  { z }  =  { <. ( 1st `  z ) ,  ( 2nd `  z
) >. } )
8864, 58xpsn 5890 . . . . . . . . . . . . 13  |-  ( { ( 1st `  z
) }  X.  {
( 2nd `  z
) } )  =  { <. ( 1st `  z
) ,  ( 2nd `  z ) >. }
8987, 88syl6eqr 2493 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  X )  ->  { z }  =  ( { ( 1st `  z
) }  X.  {
( 2nd `  z
) } ) )
9089fveq2d 5700 . . . . . . . . . . 11  |-  ( z  e.  ( X  X.  X )  ->  (
( nei `  ( J  tX  J ) ) `
 { z } )  =  ( ( nei `  ( J 
tX  J ) ) `
 ( { ( 1st `  z ) }  X.  { ( 2nd `  z ) } ) ) )
91903ad2ant3 1011 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  z  e.  ( X  X.  X
) )  ->  (
( nei `  ( J  tX  J ) ) `
 { z } )  =  ( ( nei `  ( J 
tX  J ) ) `
 ( { ( 1st `  z ) }  X.  { ( 2nd `  z ) } ) ) )
9285, 91eleqtrrd 2520 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  z  e.  ( X  X.  X
) )  ->  (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  e.  ( ( nei `  ( J  tX  J
) ) `  {
z } ) )
9375, 76, 25, 92syl3anc 1218 . . . . . . . 8  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  -> 
( ( V " { ( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  e.  ( ( nei `  ( J  tX  J
) ) `  {
z } ) )
9420neindisj 18726 . . . . . . . 8  |-  ( ( ( ( J  tX  J )  e.  Top  /\  M  C_  U. ( J  tX  J ) )  /\  ( z  e.  ( ( cls `  ( J  tX  J ) ) `
 M )  /\  ( ( V " { ( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  e.  ( ( nei `  ( J  tX  J
) ) `  {
z } ) ) )  ->  ( (
( V " {
( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M )  =/=  (/) )
957, 19, 24, 93, 94syl22anc 1219 . . . . . . 7  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  -> 
( ( ( V
" { ( 1st `  z ) } )  X.  ( V " { ( 2nd `  z
) } ) )  i^i  M )  =/=  (/) )
96 r19.3rzv 3778 . . . . . . 7  |-  ( ( ( ( V " { ( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M )  =/=  (/)  ->  ( ( 1st `  z ) ( V  o.  ( M  o.  V ) ) ( 2nd `  z )  <->  A. r  e.  (
( ( V " { ( 1st `  z
) } )  X.  ( V " {
( 2nd `  z
) } ) )  i^i  M ) ( 1st `  z ) ( V  o.  ( M  o.  V )
) ( 2nd `  z
) ) )
9795, 96syl 16 . . . . . 6  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  -> 
( ( 1st `  z
) ( V  o.  ( M  o.  V
) ) ( 2nd `  z )  <->  A. r  e.  ( ( ( V
" { ( 1st `  z ) } )  X.  ( V " { ( 2nd `  z
) } ) )  i^i  M ) ( 1st `  z ) ( V  o.  ( M  o.  V )
) ( 2nd `  z
) ) )
9874, 97mpbird 232 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  -> 
( 1st `  z
) ( V  o.  ( M  o.  V
) ) ( 2nd `  z ) )
99 df-br 4298 . . . . 5  |-  ( ( 1st `  z ) ( V  o.  ( M  o.  V )
) ( 2nd `  z
)  <->  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  e.  ( V  o.  ( M  o.  V ) ) )
10098, 99sylib 196 . . . 4  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  ->  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  e.  ( V  o.  ( M  o.  V ) ) )
10127, 100eqeltrd 2517 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X ) )  /\  ( V  e.  U  /\  `' V  =  V ) )  /\  z  e.  ( ( cls `  ( J  tX  J ) ) `  M ) )  -> 
z  e.  ( V  o.  ( M  o.  V ) ) )
102101ex 434 . 2  |-  ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X
) )  /\  ( V  e.  U  /\  `' V  =  V
) )  ->  (
z  e.  ( ( cls `  ( J 
tX  J ) ) `
 M )  -> 
z  e.  ( V  o.  ( M  o.  V ) ) ) )
103102ssrdv 3367 1  |-  ( ( ( U  e.  (UnifOn `  X )  /\  M  C_  ( X  X.  X
) )  /\  ( V  e.  U  /\  `' V  =  V
) )  ->  (
( cls `  ( J  tX  J ) ) `
 M )  C_  ( V  o.  ( M  o.  V )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   _Vcvv 2977    i^i cin 3332    C_ wss 3333   (/)c0 3642   {csn 3882   <.cop 3888   U.cuni 4096   class class class wbr 4297    X. cxp 4843   `'ccnv 4844   "cima 4848    o. ccom 4849   Rel wrel 4850   ` cfv 5423  (class class class)co 6096   1stc1st 6580   2ndc2nd 6581   Topctop 18503  TopOnctopon 18504   clsccl 18627   neicnei 18706    tX ctx 19138  UnifOncust 19779  unifTopcutop 19810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-fin 7319  df-fi 7666  df-topgen 14387  df-top 18508  df-bases 18510  df-topon 18511  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-tx 19140  df-ust 19780  df-utop 19811
This theorem is referenced by:  utopreg  19832
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