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Theorem utop2nei 18233
Description: For any symmetrical entourage  V and any relation  M, build a neighborhood of  M. First part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 14-Jan-2018.)
Hypothesis
Ref Expression
utoptop.1  |-  J  =  (unifTop `  U )
Assertion
Ref Expression
utop2nei  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  ( V  o.  ( M  o.  V
) )  e.  ( ( nei `  ( J  tX  J ) ) `
 M ) )

Proof of Theorem utop2nei
Dummy variables  r 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 utoptop.1 . . . . . . . 8  |-  J  =  (unifTop `  U )
2 utoptop 18217 . . . . . . . 8  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  e.  Top )
31, 2syl5eqel 2488 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  J  e.  Top )
4 txtop 17554 . . . . . . 7  |-  ( ( J  e.  Top  /\  J  e.  Top )  ->  ( J  tX  J
)  e.  Top )
53, 3, 4syl2anc 643 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  ( J  tX  J )  e.  Top )
653ad2ant1 978 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  ( J  tX  J )  e.  Top )
76adantr 452 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =  (/) )  ->  ( J  tX  J )  e.  Top )
8 0nei 17147 . . . 4  |-  ( ( J  tX  J )  e.  Top  ->  (/)  e.  ( ( nei `  ( J  tX  J ) ) `
 (/) ) )
97, 8syl 16 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =  (/) )  ->  (/)  e.  ( ( nei `  ( J 
tX  J ) ) `
 (/) ) )
10 coeq1 4989 . . . . . . 7  |-  ( M  =  (/)  ->  ( M  o.  V )  =  ( (/)  o.  V
) )
11 co01 5343 . . . . . . 7  |-  ( (/)  o.  V )  =  (/)
1210, 11syl6eq 2452 . . . . . 6  |-  ( M  =  (/)  ->  ( M  o.  V )  =  (/) )
1312coeq2d 4994 . . . . 5  |-  ( M  =  (/)  ->  ( V  o.  ( M  o.  V ) )  =  ( V  o.  (/) ) )
14 co02 5342 . . . . 5  |-  ( V  o.  (/) )  =  (/)
1513, 14syl6eq 2452 . . . 4  |-  ( M  =  (/)  ->  ( V  o.  ( M  o.  V ) )  =  (/) )
1615adantl 453 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =  (/) )  ->  ( V  o.  ( M  o.  V
) )  =  (/) )
17 simpr 448 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =  (/) )  ->  M  =  (/) )
1817fveq2d 5691 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =  (/) )  ->  ( ( nei `  ( J  tX  J
) ) `  M
)  =  ( ( nei `  ( J 
tX  J ) ) `
 (/) ) )
199, 16, 183eltr4d 2485 . 2  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =  (/) )  ->  ( V  o.  ( M  o.  V
) )  e.  ( ( nei `  ( J  tX  J ) ) `
 M ) )
206adantr 452 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  ( J  tX  J )  e.  Top )
21 simpl1 960 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  U  e.  (UnifOn `  X ) )
2221, 3syl 16 . . . . . . . . 9  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  J  e.  Top )
23 simpl2l 1010 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  V  e.  U )
24 simp3 959 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  M  C_  ( X  X.  X ) )
2524sselda 3308 . . . . . . . . . . 11  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  r  e.  ( X  X.  X
) )
26 xp1st 6335 . . . . . . . . . . 11  |-  ( r  e.  ( X  X.  X )  ->  ( 1st `  r )  e.  X )
2725, 26syl 16 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  ( 1st `  r )  e.  X
)
281utopsnnei 18232 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  ( 1st `  r )  e.  X )  ->  ( V " { ( 1st `  r ) } )  e.  ( ( nei `  J ) `  {
( 1st `  r
) } ) )
2921, 23, 27, 28syl3anc 1184 . . . . . . . . 9  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  ( V " { ( 1st `  r
) } )  e.  ( ( nei `  J
) `  { ( 1st `  r ) } ) )
30 xp2nd 6336 . . . . . . . . . . 11  |-  ( r  e.  ( X  X.  X )  ->  ( 2nd `  r )  e.  X )
3125, 30syl 16 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  ( 2nd `  r )  e.  X
)
321utopsnnei 18232 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  ( 2nd `  r )  e.  X )  ->  ( V " { ( 2nd `  r ) } )  e.  ( ( nei `  J ) `  {
( 2nd `  r
) } ) )
3321, 23, 31, 32syl3anc 1184 . . . . . . . . 9  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  ( V " { ( 2nd `  r
) } )  e.  ( ( nei `  J
) `  { ( 2nd `  r ) } ) )
34 eqid 2404 . . . . . . . . . 10  |-  U. J  =  U. J
3534, 34neitx 17592 . . . . . . . . 9  |-  ( ( ( J  e.  Top  /\  J  e.  Top )  /\  ( ( V " { ( 1st `  r
) } )  e.  ( ( nei `  J
) `  { ( 1st `  r ) } )  /\  ( V
" { ( 2nd `  r ) } )  e.  ( ( nei `  J ) `  {
( 2nd `  r
) } ) ) )  ->  ( ( V " { ( 1st `  r ) } )  X.  ( V " { ( 2nd `  r
) } ) )  e.  ( ( nei `  ( J  tX  J
) ) `  ( { ( 1st `  r
) }  X.  {
( 2nd `  r
) } ) ) )
3622, 22, 29, 33, 35syl22anc 1185 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  ( ( V " { ( 1st `  r ) } )  X.  ( V " { ( 2nd `  r
) } ) )  e.  ( ( nei `  ( J  tX  J
) ) `  ( { ( 1st `  r
) }  X.  {
( 2nd `  r
) } ) ) )
37 fvex 5701 . . . . . . . . . 10  |-  ( 1st `  r )  e.  _V
38 fvex 5701 . . . . . . . . . 10  |-  ( 2nd `  r )  e.  _V
3937, 38xpsn 5869 . . . . . . . . 9  |-  ( { ( 1st `  r
) }  X.  {
( 2nd `  r
) } )  =  { <. ( 1st `  r
) ,  ( 2nd `  r ) >. }
4039fveq2i 5690 . . . . . . . 8  |-  ( ( nei `  ( J 
tX  J ) ) `
 ( { ( 1st `  r ) }  X.  { ( 2nd `  r ) } ) )  =  ( ( nei `  ( J  tX  J ) ) `
 { <. ( 1st `  r ) ,  ( 2nd `  r
) >. } )
4136, 40syl6eleq 2494 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  ( ( V " { ( 1st `  r ) } )  X.  ( V " { ( 2nd `  r
) } ) )  e.  ( ( nei `  ( J  tX  J
) ) `  { <. ( 1st `  r
) ,  ( 2nd `  r ) >. } ) )
4224adantr 452 . . . . . . . . . . 11  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  M  C_  ( X  X.  X ) )
43 xpss 4941 . . . . . . . . . . . . 13  |-  ( X  X.  X )  C_  ( _V  X.  _V )
44 sstr 3316 . . . . . . . . . . . . 13  |-  ( ( M  C_  ( X  X.  X )  /\  ( X  X.  X )  C_  ( _V  X.  _V )
)  ->  M  C_  ( _V  X.  _V ) )
4543, 44mpan2 653 . . . . . . . . . . . 12  |-  ( M 
C_  ( X  X.  X )  ->  M  C_  ( _V  X.  _V ) )
46 df-rel 4844 . . . . . . . . . . . 12  |-  ( Rel 
M  <->  M  C_  ( _V 
X.  _V ) )
4745, 46sylibr 204 . . . . . . . . . . 11  |-  ( M 
C_  ( X  X.  X )  ->  Rel  M )
4842, 47syl 16 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  Rel  M )
49 1st2nd 6352 . . . . . . . . . 10  |-  ( ( Rel  M  /\  r  e.  M )  ->  r  =  <. ( 1st `  r
) ,  ( 2nd `  r ) >. )
5048, 49sylancom 649 . . . . . . . . 9  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  r  =  <. ( 1st `  r
) ,  ( 2nd `  r ) >. )
5150sneqd 3787 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  { r }  =  { <. ( 1st `  r ) ,  ( 2nd `  r
) >. } )
5251fveq2d 5691 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  ( ( nei `  ( J  tX  J ) ) `  { r } )  =  ( ( nei `  ( J  tX  J
) ) `  { <. ( 1st `  r
) ,  ( 2nd `  r ) >. } ) )
5341, 52eleqtrrd 2481 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  ( ( V " { ( 1st `  r ) } )  X.  ( V " { ( 2nd `  r
) } ) )  e.  ( ( nei `  ( J  tX  J
) ) `  {
r } ) )
54 relxp 4942 . . . . . . . . . . 11  |-  Rel  (
( V " {
( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) )
5554a1i 11 . . . . . . . . . 10  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  Rel  ( ( V " { ( 1st `  r ) } )  X.  ( V " { ( 2nd `  r ) } ) ) )
56 1st2nd 6352 . . . . . . . . . 10  |-  ( ( Rel  ( ( V
" { ( 1st `  r ) } )  X.  ( V " { ( 2nd `  r
) } ) )  /\  z  e.  ( ( V " {
( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
5755, 56sylancom 649 . . . . . . . . 9  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
58 simpll2 997 . . . . . . . . . . . . 13  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  ( V  e.  U  /\  `' V  =  V ) )
5958simprd 450 . . . . . . . . . . . 12  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  `' V  =  V )
60 simpll1 996 . . . . . . . . . . . . . 14  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  U  e.  (UnifOn `  X ) )
6158simpld 446 . . . . . . . . . . . . . 14  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  V  e.  U )
62 ustrel 18194 . . . . . . . . . . . . . 14  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  Rel  V )
6360, 61, 62syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  Rel  V )
64 xp1st 6335 . . . . . . . . . . . . . 14  |-  ( z  e.  ( ( V
" { ( 1st `  r ) } )  X.  ( V " { ( 2nd `  r
) } ) )  ->  ( 1st `  z
)  e.  ( V
" { ( 1st `  r ) } ) )
6564adantl 453 . . . . . . . . . . . . 13  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  ( 1st `  z )  e.  ( V " { ( 1st `  r ) } ) )
66 elrelimasn 5187 . . . . . . . . . . . . . 14  |-  ( Rel 
V  ->  ( ( 1st `  z )  e.  ( V " {
( 1st `  r
) } )  <->  ( 1st `  r ) V ( 1st `  z ) ) )
6766biimpa 471 . . . . . . . . . . . . 13  |-  ( ( Rel  V  /\  ( 1st `  z )  e.  ( V " {
( 1st `  r
) } ) )  ->  ( 1st `  r
) V ( 1st `  z ) )
6863, 65, 67syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  ( 1st `  r ) V ( 1st `  z ) )
69 fvex 5701 . . . . . . . . . . . . . . 15  |-  ( 1st `  z )  e.  _V
7037, 69brcnv 5014 . . . . . . . . . . . . . 14  |-  ( ( 1st `  r ) `' V ( 1st `  z
)  <->  ( 1st `  z
) V ( 1st `  r ) )
71 breq 4174 . . . . . . . . . . . . . 14  |-  ( `' V  =  V  -> 
( ( 1st `  r
) `' V ( 1st `  z )  <-> 
( 1st `  r
) V ( 1st `  z ) ) )
7270, 71syl5bbr 251 . . . . . . . . . . . . 13  |-  ( `' V  =  V  -> 
( ( 1st `  z
) V ( 1st `  r )  <->  ( 1st `  r ) V ( 1st `  z ) ) )
7372biimpar 472 . . . . . . . . . . . 12  |-  ( ( `' V  =  V  /\  ( 1st `  r
) V ( 1st `  z ) )  -> 
( 1st `  z
) V ( 1st `  r ) )
7459, 68, 73syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  ( 1st `  z ) V ( 1st `  r ) )
75 simpll3 998 . . . . . . . . . . . 12  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  M  C_  ( X  X.  X ) )
76 simplr 732 . . . . . . . . . . . 12  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  r  e.  M )
77 1st2ndbr 6355 . . . . . . . . . . . . 13  |-  ( ( Rel  M  /\  r  e.  M )  ->  ( 1st `  r ) M ( 2nd `  r
) )
7847, 77sylan 458 . . . . . . . . . . . 12  |-  ( ( M  C_  ( X  X.  X )  /\  r  e.  M )  ->  ( 1st `  r ) M ( 2nd `  r
) )
7975, 76, 78syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  ( 1st `  r ) M ( 2nd `  r ) )
80 xp2nd 6336 . . . . . . . . . . . . 13  |-  ( z  e.  ( ( V
" { ( 1st `  r ) } )  X.  ( V " { ( 2nd `  r
) } ) )  ->  ( 2nd `  z
)  e.  ( V
" { ( 2nd `  r ) } ) )
8180adantl 453 . . . . . . . . . . . 12  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  ( 2nd `  z )  e.  ( V " { ( 2nd `  r ) } ) )
82 elrelimasn 5187 . . . . . . . . . . . . 13  |-  ( Rel 
V  ->  ( ( 2nd `  z )  e.  ( V " {
( 2nd `  r
) } )  <->  ( 2nd `  r ) V ( 2nd `  z ) ) )
8382biimpa 471 . . . . . . . . . . . 12  |-  ( ( Rel  V  /\  ( 2nd `  z )  e.  ( V " {
( 2nd `  r
) } ) )  ->  ( 2nd `  r
) V ( 2nd `  z ) )
8463, 81, 83syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  ( 2nd `  r ) V ( 2nd `  z ) )
8569, 38, 373pm3.2i 1132 . . . . . . . . . . . . 13  |-  ( ( 1st `  z )  e.  _V  /\  ( 2nd `  r )  e. 
_V  /\  ( 1st `  r )  e.  _V )
86 brcogw 5000 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 ) )
8785, 86mpan 652 . . . . . . . . . . . 12  |-  ( ( ( 1st `  z
) V ( 1st `  r )  /\  ( 1st `  r ) M ( 2nd `  r
) )  ->  ( 1st `  z ) ( M  o.  V ) ( 2nd `  r
) )
88 fvex 5701 . . . . . . . . . . . . . 14  |-  ( 2nd `  z )  e.  _V
8969, 88, 383pm3.2i 1132 . . . . . . . . . . . . 13  |-  ( ( 1st `  z )  e.  _V  /\  ( 2nd `  z )  e. 
_V  /\  ( 2nd `  r )  e.  _V )
90 brcogw 5000 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 ) )
9189, 90mpan 652 . . . . . . . . . . . 12  |-  ( ( ( 1st `  z
) ( M  o.  V ) ( 2nd `  r )  /\  ( 2nd `  r ) V ( 2nd `  z
) )  ->  ( 1st `  z ) ( V  o.  ( M  o.  V ) ) ( 2nd `  z
) )
9287, 91sylan 458 . . . . . . . . . . 11  |-  ( ( ( ( 1st `  z
) V ( 1st `  r )  /\  ( 1st `  r ) M ( 2nd `  r
) )  /\  ( 2nd `  r ) V ( 2nd `  z
) )  ->  ( 1st `  z ) ( V  o.  ( M  o.  V ) ) ( 2nd `  z
) )
9374, 79, 84, 92syl21anc 1183 . . . . . . . . . 10  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  ( 1st `  z ) ( V  o.  ( M  o.  V ) ) ( 2nd `  z ) )
94 df-br 4173 . . . . . . . . . 10  |-  ( ( 1st `  z ) ( V  o.  ( M  o.  V )
) ( 2nd `  z
)  <->  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  e.  ( V  o.  ( M  o.  V ) ) )
9593, 94sylib 189 . . . . . . . . 9  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  <. ( 1st `  z ) ,  ( 2nd `  z )
>.  e.  ( V  o.  ( M  o.  V
) ) )
9657, 95eqeltrd 2478 . . . . . . . 8  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  /\  z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) ) )  ->  z  e.  ( V  o.  ( M  o.  V )
) )
9796ex 424 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  ( z  e.  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) )  ->  z  e.  ( V  o.  ( M  o.  V ) ) ) )
9897ssrdv 3314 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  ( ( V " { ( 1st `  r ) } )  X.  ( V " { ( 2nd `  r
) } ) ) 
C_  ( V  o.  ( M  o.  V
) ) )
99 simp1 957 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  U  e.  (UnifOn `  X ) )
100 simp2l 983 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  V  e.  U
)
101 ustssxp 18187 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( X  X.  X
) )
10299, 100, 101syl2anc 643 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  V  C_  ( X  X.  X ) )
103 coss1 4987 . . . . . . . . . 10  |-  ( V 
C_  ( X  X.  X )  ->  ( V  o.  ( M  o.  V ) )  C_  ( ( X  X.  X )  o.  ( M  o.  V )
) )
104102, 103syl 16 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  ( V  o.  ( M  o.  V
) )  C_  (
( X  X.  X
)  o.  ( M  o.  V ) ) )
105 coss1 4987 . . . . . . . . . . . 12  |-  ( M 
C_  ( X  X.  X )  ->  ( M  o.  V )  C_  ( ( X  X.  X )  o.  V
) )
10624, 105syl 16 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  ( M  o.  V )  C_  (
( X  X.  X
)  o.  V ) )
107 coss2 4988 . . . . . . . . . . . . 13  |-  ( V 
C_  ( X  X.  X )  ->  (
( X  X.  X
)  o.  V ) 
C_  ( ( X  X.  X )  o.  ( X  X.  X
) ) )
108 xpcoid 5374 . . . . . . . . . . . . 13  |-  ( ( X  X.  X )  o.  ( X  X.  X ) )  =  ( X  X.  X
)
109107, 108syl6sseq 3354 . . . . . . . . . . . 12  |-  ( V 
C_  ( X  X.  X )  ->  (
( X  X.  X
)  o.  V ) 
C_  ( X  X.  X ) )
110102, 109syl 16 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  ( ( X  X.  X )  o.  V )  C_  ( X  X.  X ) )
111106, 110sstrd 3318 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  ( M  o.  V )  C_  ( X  X.  X ) )
112 coss2 4988 . . . . . . . . . . 11  |-  ( ( M  o.  V ) 
C_  ( X  X.  X )  ->  (
( X  X.  X
)  o.  ( M  o.  V ) ) 
C_  ( ( X  X.  X )  o.  ( X  X.  X
) ) )
113112, 108syl6sseq 3354 . . . . . . . . . 10  |-  ( ( M  o.  V ) 
C_  ( X  X.  X )  ->  (
( X  X.  X
)  o.  ( M  o.  V ) ) 
C_  ( X  X.  X ) )
114111, 113syl 16 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  ( ( X  X.  X )  o.  ( M  o.  V
) )  C_  ( X  X.  X ) )
115104, 114sstrd 3318 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  ( V  o.  ( M  o.  V
) )  C_  ( X  X.  X ) )
116 utopbas 18218 . . . . . . . . . . . 12  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. (unifTop `  U )
)
1171unieqi 3985 . . . . . . . . . . . 12  |-  U. J  =  U. (unifTop `  U
)
118116, 117syl6eqr 2454 . . . . . . . . . . 11  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. J )
119118, 118xpeq12d 4862 . . . . . . . . . 10  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  =  ( U. J  X.  U. J ) )
12034, 34txuni 17577 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  J  e.  Top )  ->  ( U. J  X.  U. J )  =  U. ( J  tX  J ) )
1213, 3, 120syl2anc 643 . . . . . . . . . 10  |-  ( U  e.  (UnifOn `  X
)  ->  ( U. J  X.  U. J )  =  U. ( J 
tX  J ) )
122119, 121eqtrd 2436 . . . . . . . . 9  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  =  U. ( J  tX  J ) )
1231223ad2ant1 978 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  ( X  X.  X )  =  U. ( J  tX  J ) )
124115, 123sseqtrd 3344 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  ( V  o.  ( M  o.  V
) )  C_  U. ( J  tX  J ) )
125124adantr 452 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  ( V  o.  ( M  o.  V
) )  C_  U. ( J  tX  J ) )
126 eqid 2404 . . . . . . 7  |-  U. ( J  tX  J )  = 
U. ( J  tX  J )
127126ssnei2 17135 . . . . . 6  |-  ( ( ( ( J  tX  J )  e.  Top  /\  ( ( V " { ( 1st `  r
) } )  X.  ( V " {
( 2nd `  r
) } ) )  e.  ( ( nei `  ( J  tX  J
) ) `  {
r } ) )  /\  ( ( ( V " { ( 1st `  r ) } )  X.  ( V " { ( 2nd `  r ) } ) )  C_  ( V  o.  ( M  o.  V
) )  /\  ( V  o.  ( M  o.  V ) )  C_  U. ( J  tX  J
) ) )  -> 
( V  o.  ( M  o.  V )
)  e.  ( ( nei `  ( J 
tX  J ) ) `
 { r } ) )
12820, 53, 98, 125, 127syl22anc 1185 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  ( V  o.  ( M  o.  V
) )  e.  ( ( nei `  ( J  tX  J ) ) `
 { r } ) )
129128ralrimiva 2749 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  A. r  e.  M  ( V  o.  ( M  o.  V )
)  e.  ( ( nei `  ( J 
tX  J ) ) `
 { r } ) )
130129adantr 452 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =/=  (/) )  ->  A. r  e.  M  ( V  o.  ( M  o.  V )
)  e.  ( ( nei `  ( J 
tX  J ) ) `
 { r } ) )
1316adantr 452 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =/=  (/) )  -> 
( J  tX  J
)  e.  Top )
13224, 123sseqtrd 3344 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  M  C_  U. ( J  tX  J ) )
133132adantr 452 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =/=  (/) )  ->  M  C_  U. ( J 
tX  J ) )
134 simpr 448 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =/=  (/) )  ->  M  =/=  (/) )
135126neips 17132 . . . 4  |-  ( ( ( J  tX  J
)  e.  Top  /\  M  C_  U. ( J 
tX  J )  /\  M  =/=  (/) )  ->  (
( V  o.  ( M  o.  V )
)  e.  ( ( nei `  ( J 
tX  J ) ) `
 M )  <->  A. r  e.  M  ( V  o.  ( M  o.  V
) )  e.  ( ( nei `  ( J  tX  J ) ) `
 { r } ) ) )
136131, 133, 134, 135syl3anc 1184 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =/=  (/) )  -> 
( ( V  o.  ( M  o.  V
) )  e.  ( ( nei `  ( J  tX  J ) ) `
 M )  <->  A. r  e.  M  ( V  o.  ( M  o.  V
) )  e.  ( ( nei `  ( J  tX  J ) ) `
 { r } ) ) )
137130, 136mpbird 224 . 2  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =/=  (/) )  -> 
( V  o.  ( M  o.  V )
)  e.  ( ( nei `  ( J 
tX  J ) ) `
 M ) )
13819, 137pm2.61dane 2645 1  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  ( V  o.  ( M  o.  V
) )  e.  ( ( nei `  ( J  tX  J ) ) `
 M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   _Vcvv 2916    C_ wss 3280   (/)c0 3588   {csn 3774   <.cop 3777   U.cuni 3975   class class class wbr 4172    X. cxp 4835   `'ccnv 4836   "cima 4840    o. ccom 4841   Rel wrel 4842   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   Topctop 16913   neicnei 17116    tX ctx 17545  UnifOncust 18182  unifTopcutop 18213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-fin 7072  df-fi 7374  df-topgen 13622  df-top 16918  df-bases 16920  df-topon 16921  df-nei 17117  df-tx 17547  df-ust 18183  df-utop 18214
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