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Theorem utop3cls 18269
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 4974 . . . . 5  |-  Rel  ( X  X.  X )
2 utoptop.1 . . . . . . . . . . 11  |-  J  =  (unifTop `  U )
3 utoptop 18252 . . . . . . . . . . 11  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  e.  Top )
42, 3syl5eqel 2519 . . . . . . . . . 10  |-  ( U  e.  (UnifOn `  X
)  ->  J  e.  Top )
5 txtop 17589 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  J  e.  Top )  ->  ( J  tX  J
)  e.  Top )
64, 4, 5syl2anc 643 . . . . . . . . 9  |-  ( U  e.  (UnifOn `  X
)  ->  ( J  tX  J )  e.  Top )
76ad3antrrr 711 . . . . . . . 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 736 . . . . . . . . 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 18254 . . . . . . . . . . . . . 14  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  e.  (TopOn `  X ) )
102, 9syl5eqel 2519 . . . . . . . . . . . . 13  |-  ( U  e.  (UnifOn `  X
)  ->  J  e.  (TopOn `  X ) )
11 toponuni 16980 . . . . . . . . . . . . 13  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1210, 11syl 16 . . . . . . . . . . . 12  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. J )
1312, 12xpeq12d 4894 . . . . . . . . . . 11  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  =  ( U. J  X.  U. J ) )
14 eqid 2435 . . . . . . . . . . . . 13  |-  U. J  =  U. J
1514, 14txuni 17612 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  J  e.  Top )  ->  ( U. J  X.  U. J )  =  U. ( J  tX  J ) )
164, 4, 15syl2anc 643 . . . . . . . . . . 11  |-  ( U  e.  (UnifOn `  X
)  ->  ( U. J  X.  U. J )  =  U. ( J 
tX  J ) )
1713, 16eqtrd 2467 . . . . . . . . . 10  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  =  U. ( J  tX  J ) )
1817ad3antrrr 711 . . . . . . . . 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 3376 . . . . . . . 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 2435 . . . . . . . . 9  |-  U. ( J  tX  J )  = 
U. ( J  tX  J )
2120clsss3 17111 . . . . . . . 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 643 . . . . . . 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 3377 . . . . . 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 448 . . . . . 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 3341 . . . . 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 6384 . . . . 5  |-  ( ( Rel  ( X  X.  X )  /\  z  e.  ( X  X.  X
) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
271, 25, 26sylancr 645 . . . 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 743 . . . . . . . . . 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 1014 . . . . . . . . . . 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 1175 . . . . . . . . . 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 18229 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  Rel  V )
3228, 30, 31syl2anc 643 . . . . . . . . 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 448 . . . . . . . . . . . 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 3522 . . . . . . . . . . . 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 189 . . . . . . . . . . 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 446 . . . . . . . . . 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 6367 . . . . . . . . . 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 5219 . . . . . . . . . 10  |-  ( Rel 
V  ->  ( ( 1st `  r )  e.  ( V " {
( 1st `  z
) } )  <->  ( 1st `  z ) V ( 1st `  r ) ) )
4039biimpa 471 . . . . . . . . 9  |-  ( ( Rel  V  /\  ( 1st `  r )  e.  ( V " {
( 1st `  z
) } ) )  ->  ( 1st `  z
) V ( 1st `  r ) )
4132, 38, 40syl2anc 643 . . . . . . . 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 744 . . . . . . . . . . 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 4973 . . . . . . . . . . 11  |-  ( X  X.  X )  C_  ( _V  X.  _V )
4442, 43syl6ss 3352 . . . . . . . . . 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 4876 . . . . . . . . . 10  |-  ( Rel 
M  <->  M  C_  ( _V 
X.  _V ) )
4644, 45sylibr 204 . . . . . . . . 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 450 . . . . . . . . 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 6387 . . . . . . . . 9  |-  ( ( Rel  M  /\  r  e.  M )  ->  ( 1st `  r ) M ( 2nd `  r
) )
4946, 47, 48syl2anc 643 . . . . . . . 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 6368 . . . . . . . . . . 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 5219 . . . . . . . . . . 11  |-  ( Rel 
V  ->  ( ( 2nd `  r )  e.  ( V " {
( 2nd `  z
) } )  <->  ( 2nd `  z ) V ( 2nd `  r ) ) )
5352biimpa 471 . . . . . . . . . 10  |-  ( ( Rel  V  /\  ( 2nd `  r )  e.  ( V " {
( 2nd `  z
) } ) )  ->  ( 2nd `  z
) V ( 2nd `  r ) )
5432, 51, 53syl2anc 643 . . . . . . . . 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 1015 . . . . . . . . . . 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 1175 . . . . . . . . . 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 5733 . . . . . . . . . . . 12  |-  ( 2nd `  r )  e.  _V
58 fvex 5733 . . . . . . . . . . . 12  |-  ( 2nd `  z )  e.  _V
5957, 58brcnv 5046 . . . . . . . . . . 11  |-  ( ( 2nd `  r ) `' V ( 2nd `  z
)  <->  ( 2nd `  z
) V ( 2nd `  r ) )
60 breq 4206 . . . . . . . . . . 11  |-  ( `' V  =  V  -> 
( ( 2nd `  r
) `' V ( 2nd `  z )  <-> 
( 2nd `  r
) V ( 2nd `  z ) ) )
6159, 60syl5rbbr 252 . . . . . . . . . 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 224 . . . . . . . 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 5733 . . . . . . . . . 10  |-  ( 1st `  z )  e.  _V
65 fvex 5733 . . . . . . . . . 10  |-  ( 1st `  r )  e.  _V
66 brcogw 5032 . . . . . . . . . . 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 424 . . . . . . . . . 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 1279 . . . . . . . . 9  |-  ( ( ( 1st `  z
) V ( 1st `  r )  /\  ( 1st `  r ) M ( 2nd `  r
) )  ->  ( 1st `  z ) ( M  o.  V ) ( 2nd `  r
) )
69 brcogw 5032 . . . . . . . . . . 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 424 . . . . . . . . . 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 1279 . . . . . . . . 9  |-  ( ( ( 1st `  z
) ( M  o.  V ) ( 2nd `  r )  /\  ( 2nd `  r ) V ( 2nd `  z
) )  ->  ( 1st `  z ) ( V  o.  ( M  o.  V ) ) ( 2nd `  z
) )
7268, 71sylan 458 . . . . . . . 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 1183 . . . . . . 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 2781 . . . . . 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 735 . . . . . . . . 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 737 . . . . . . . . 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 978 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  z  e.  ( X  X.  X
) )  ->  J  e.  Top )
78 xp1st 6367 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  X )  ->  ( 1st `  z )  e.  X )
792utopsnnei 18267 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  ( 1st `  z )  e.  X )  ->  ( V " { ( 1st `  z ) } )  e.  ( ( nei `  J ) `  {
( 1st `  z
) } ) )
8078, 79syl3an3 1219 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  z  e.  ( X  X.  X
) )  ->  ( V " { ( 1st `  z ) } )  e.  ( ( nei `  J ) `  {
( 1st `  z
) } ) )
81 xp2nd 6368 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  X )  ->  ( 2nd `  z )  e.  X )
822utopsnnei 18267 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  ( 2nd `  z )  e.  X )  ->  ( V " { ( 2nd `  z ) } )  e.  ( ( nei `  J ) `  {
( 2nd `  z
) } ) )
8381, 82syl3an3 1219 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  z  e.  ( X  X.  X
) )  ->  ( V " { ( 2nd `  z ) } )  e.  ( ( nei `  J ) `  {
( 2nd `  z
) } ) )
8414, 14neitx 17627 . . . . . . . . . . 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 1185 . . . . . . . . . 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 6377 . . . . . . . . . . . . . 14  |-  ( z  e.  ( X  X.  X )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
8786sneqd 3819 . . . . . . . . . . . . 13  |-  ( z  e.  ( X  X.  X )  ->  { z }  =  { <. ( 1st `  z ) ,  ( 2nd `  z
) >. } )
8864, 58xpsn 5901 . . . . . . . . . . . . 13  |-  ( { ( 1st `  z
) }  X.  {
( 2nd `  z
) } )  =  { <. ( 1st `  z
) ,  ( 2nd `  z ) >. }
8987, 88syl6eqr 2485 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  X )  ->  { z }  =  ( { ( 1st `  z
) }  X.  {
( 2nd `  z
) } ) )
9089fveq2d 5723 . . . . . . . . . . 11  |-  ( z  e.  ( X  X.  X )  ->  (
( nei `  ( J  tX  J ) ) `
 { z } )  =  ( ( nei `  ( J 
tX  J ) ) `
 ( { ( 1st `  z ) }  X.  { ( 2nd `  z ) } ) ) )
91903ad2ant3 980 . . . . . . . . . 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 2512 . . . . . . . . 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 1184 . . . . . . . 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 17169 . . . . . . . 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 1185 . . . . . . 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 3713 . . . . . . 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 224 . . . . 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 4205 . . . . 5  |-  ( ( 1st `  z ) ( V  o.  ( M  o.  V )
) ( 2nd `  z
)  <->  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  e.  ( V  o.  ( M  o.  V ) ) )
10098, 99sylib 189 . . . 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 2509 . . 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 424 . 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 3346 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   _Vcvv 2948    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806   <.cop 3809   U.cuni 4007   class class class wbr 4204    X. cxp 4867   `'ccnv 4868   "cima 4872    o. ccom 4873   Rel wrel 4874   ` cfv 5445  (class class class)co 6072   1stc1st 6338   2ndc2nd 6339   Topctop 16946  TopOnctopon 16947   clsccl 17070   neicnei 17149    tX ctx 17580  UnifOncust 18217  unifTopcutop 18248
This theorem is referenced by:  utopreg  18270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-en 7101  df-fin 7104  df-fi 7407  df-topgen 13655  df-top 16951  df-bases 16953  df-topon 16954  df-cld 17071  df-ntr 17072  df-cls 17073  df-nei 17150  df-tx 17582  df-ust 18218  df-utop 18249
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