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Theorem utop3cls 20880
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 5119 . . . . 5  |-  Rel  ( X  X.  X )
2 utoptop.1 . . . . . . . . . . 11  |-  J  =  (unifTop `  U )
3 utoptop 20863 . . . . . . . . . . 11  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  e.  Top )
42, 3syl5eqel 2549 . . . . . . . . . 10  |-  ( U  e.  (UnifOn `  X
)  ->  J  e.  Top )
5 txtop 20196 . . . . . . . . . 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 760 . . . . . . . . 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 20865 . . . . . . . . . . . . . 14  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  e.  (TopOn `  X ) )
102, 9syl5eqel 2549 . . . . . . . . . . . . 13  |-  ( U  e.  (UnifOn `  X
)  ->  J  e.  (TopOn `  X ) )
11 toponuni 19555 . . . . . . . . . . . . 13  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1210, 11syl 16 . . . . . . . . . . . 12  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. J )
1312sqxpeqd 5034 . . . . . . . . . . 11  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  =  ( U. J  X.  U. J ) )
14 eqid 2457 . . . . . . . . . . . . 13  |-  U. J  =  U. J
1514, 14txuni 20219 . . . . . . . . . . . 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 2498 . . . . . . . . . 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 3535 . . . . . . . 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 2457 . . . . . . . . 9  |-  U. ( J  tX  J )  = 
U. ( J  tX  J )
2120clsss3 19687 . . . . . . . 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 3536 . . . . . 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 3500 . . . . 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 6845 . . . . 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 767 . . . . . . . . . 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 20840 . . . . . . . . . 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 3683 . . . . . . . . . . . 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 6829 . . . . . . . . . 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 5371 . . . . . . . . . 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 768 . . . . . . . . . . 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 5118 . . . . . . . . . . 11  |-  ( X  X.  X )  C_  ( _V  X.  _V )
4442, 43syl6ss 3511 . . . . . . . . . 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 5015 . . . . . . . . . 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 6848 . . . . . . . . 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 6830 . . . . . . . . . . 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 5371 . . . . . . . . . . 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 5882 . . . . . . . . . . . 12  |-  ( 2nd `  r )  e.  _V
58 fvex 5882 . . . . . . . . . . . 12  |-  ( 2nd `  z )  e.  _V
5957, 58brcnv 5195 . . . . . . . . . . 11  |-  ( ( 2nd `  r ) `' V ( 2nd `  z
)  <->  ( 2nd `  z
) V ( 2nd `  r ) )
60 breq 4458 . . . . . . . . . . 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 5882 . . . . . . . . . 10  |-  ( 1st `  z )  e.  _V
65 fvex 5882 . . . . . . . . . 10  |-  ( 1st `  r )  e.  _V
66 brcogw 5181 . . . . . . . . . . 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 5181 . . . . . . . . . . 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 2871 . . . . . 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 759 . . . . . . . . 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 761 . . . . . . . . 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 6829 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  X )  ->  ( 1st `  z )  e.  X )
792utopsnnei 20878 . . . . . . . . . . . 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 6830 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  X )  ->  ( 2nd `  z )  e.  X )
822utopsnnei 20878 . . . . . . . . . . . 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 20234 . . . . . . . . . . 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 6836 . . . . . . . . . . . . . 14  |-  ( z  e.  ( X  X.  X )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
8786sneqd 4044 . . . . . . . . . . . . 13  |-  ( z  e.  ( X  X.  X )  ->  { z }  =  { <. ( 1st `  z ) ,  ( 2nd `  z
) >. } )
8864, 58xpsn 6074 . . . . . . . . . . . . 13  |-  ( { ( 1st `  z
) }  X.  {
( 2nd `  z
) } )  =  { <. ( 1st `  z
) ,  ( 2nd `  z ) >. }
8987, 88syl6eqr 2516 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  X )  ->  { z }  =  ( { ( 1st `  z
) }  X.  {
( 2nd `  z
) } ) )
9089fveq2d 5876 . . . . . . . . . . 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 2548 . . . . . . . . 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 19745 . . . . . . . 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 3925 . . . . . . 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 4457 . . . . 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 2545 . . 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 3505 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   {csn 4032   <.cop 4038   U.cuni 4251   class class class wbr 4456    X. cxp 5006   `'ccnv 5007   "cima 5011    o. ccom 5012   Rel wrel 5013   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   Topctop 19521  TopOnctopon 19522   clsccl 19646   neicnei 19725    tX ctx 20187  UnifOncust 20828  unifTopcutop 20859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-fin 7539  df-fi 7889  df-topgen 14861  df-top 19526  df-bases 19528  df-topon 19529  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-tx 20189  df-ust 20829  df-utop 20860
This theorem is referenced by:  utopreg  20881
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