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Theorem utop3cls 20489
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 5108 . . . . 5  |-  Rel  ( X  X.  X )
2 utoptop.1 . . . . . . . . . . 11  |-  J  =  (unifTop `  U )
3 utoptop 20472 . . . . . . . . . . 11  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  e.  Top )
42, 3syl5eqel 2559 . . . . . . . . . 10  |-  ( U  e.  (UnifOn `  X
)  ->  J  e.  Top )
5 txtop 19805 . . . . . . . . . 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 20474 . . . . . . . . . . . . . 14  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  e.  (TopOn `  X ) )
102, 9syl5eqel 2559 . . . . . . . . . . . . 13  |-  ( U  e.  (UnifOn `  X
)  ->  J  e.  (TopOn `  X ) )
11 toponuni 19195 . . . . . . . . . . . . 13  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1210, 11syl 16 . . . . . . . . . . . 12  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. J )
1312, 12xpeq12d 5024 . . . . . . . . . . 11  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  =  ( U. J  X.  U. J ) )
14 eqid 2467 . . . . . . . . . . . . 13  |-  U. J  =  U. J
1514, 14txuni 19828 . . . . . . . . . . . 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 2508 . . . . . . . . . 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 3540 . . . . . . . 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 2467 . . . . . . . . 9  |-  U. ( J  tX  J )  = 
U. ( J  tX  J )
2120clsss3 19326 . . . . . . . 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 3541 . . . . . 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 3505 . . . . 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 6827 . . . . 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 1053 . . . . . . . . . . 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 1218 . . . . . . . . . 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 20449 . . . . . . . . . 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 3687 . . . . . . . . . . . 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 6811 . . . . . . . . . 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 5359 . . . . . . . . . 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 5107 . . . . . . . . . . 11  |-  ( X  X.  X )  C_  ( _V  X.  _V )
4442, 43syl6ss 3516 . . . . . . . . . 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 5006 . . . . . . . . . 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 6830 . . . . . . . . 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 6812 . . . . . . . . . . 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 5359 . . . . . . . . . . 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 1054 . . . . . . . . . . 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 1218 . . . . . . . . . 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 5874 . . . . . . . . . . . 12  |-  ( 2nd `  r )  e.  _V
58 fvex 5874 . . . . . . . . . . . 12  |-  ( 2nd `  z )  e.  _V
5957, 58brcnv 5183 . . . . . . . . . . 11  |-  ( ( 2nd `  r ) `' V ( 2nd `  z
)  <->  ( 2nd `  z
) V ( 2nd `  r ) )
60 breq 4449 . . . . . . . . . . 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 5874 . . . . . . . . . 10  |-  ( 1st `  z )  e.  _V
65 fvex 5874 . . . . . . . . . 10  |-  ( 1st `  r )  e.  _V
66 brcogw 5169 . . . . . . . . . . 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 1324 . . . . . . . . 9  |-  ( ( ( 1st `  z
) V ( 1st `  r )  /\  ( 1st `  r ) M ( 2nd `  r
) )  ->  ( 1st `  z ) ( M  o.  V ) ( 2nd `  r
) )
69 brcogw 5169 . . . . . . . . . . 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 1324 . . . . . . . . 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 1227 . . . . . . 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 2878 . . . . . 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 1017 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  z  e.  ( X  X.  X
) )  ->  J  e.  Top )
78 xp1st 6811 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  X )  ->  ( 1st `  z )  e.  X )
792utopsnnei 20487 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  ( 1st `  z )  e.  X )  ->  ( V " { ( 1st `  z ) } )  e.  ( ( nei `  J ) `  {
( 1st `  z
) } ) )
8078, 79syl3an3 1263 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  z  e.  ( X  X.  X
) )  ->  ( V " { ( 1st `  z ) } )  e.  ( ( nei `  J ) `  {
( 1st `  z
) } ) )
81 xp2nd 6812 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  X )  ->  ( 2nd `  z )  e.  X )
822utopsnnei 20487 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  ( 2nd `  z )  e.  X )  ->  ( V " { ( 2nd `  z ) } )  e.  ( ( nei `  J ) `  {
( 2nd `  z
) } ) )
8381, 82syl3an3 1263 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  z  e.  ( X  X.  X
) )  ->  ( V " { ( 2nd `  z ) } )  e.  ( ( nei `  J ) `  {
( 2nd `  z
) } ) )
8414, 14neitx 19843 . . . . . . . . . . 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 1229 . . . . . . . . . 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 6818 . . . . . . . . . . . . . 14  |-  ( z  e.  ( X  X.  X )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
8786sneqd 4039 . . . . . . . . . . . . 13  |-  ( z  e.  ( X  X.  X )  ->  { z }  =  { <. ( 1st `  z ) ,  ( 2nd `  z
) >. } )
8864, 58xpsn 6061 . . . . . . . . . . . . 13  |-  ( { ( 1st `  z
) }  X.  {
( 2nd `  z
) } )  =  { <. ( 1st `  z
) ,  ( 2nd `  z ) >. }
8987, 88syl6eqr 2526 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  X )  ->  { z }  =  ( { ( 1st `  z
) }  X.  {
( 2nd `  z
) } ) )
9089fveq2d 5868 . . . . . . . . . . 11  |-  ( z  e.  ( X  X.  X )  ->  (
( nei `  ( J  tX  J ) ) `
 { z } )  =  ( ( nei `  ( J 
tX  J ) ) `
 ( { ( 1st `  z ) }  X.  { ( 2nd `  z ) } ) ) )
91903ad2ant3 1019 . . . . . . . . . 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 2558 . . . . . . . . 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 1228 . . . . . . . 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 19384 . . . . . . . 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 1229 . . . . . . 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 3921 . . . . . . 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 4448 . . . . 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 2555 . . 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 3510 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   U.cuni 4245   class class class wbr 4447    X. cxp 4997   `'ccnv 4998   "cima 5002    o. ccom 5003   Rel wrel 5004   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   Topctop 19161  TopOnctopon 19162   clsccl 19285   neicnei 19364    tX ctx 19796  UnifOncust 20437  unifTopcutop 20468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-fin 7517  df-fi 7867  df-topgen 14695  df-top 19166  df-bases 19168  df-topon 19169  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-tx 19798  df-ust 20438  df-utop 20469
This theorem is referenced by:  utopreg  20490
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