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Theorem utop2nei 21343
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 21327 . . . . . . . 8  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  e.  Top )
31, 2syl5eqel 2553 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  J  e.  Top )
4 txtop 20661 . . . . . . 7  |-  ( ( J  e.  Top  /\  J  e.  Top )  ->  ( J  tX  J
)  e.  Top )
53, 3, 4syl2anc 673 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  ( J  tX  J )  e.  Top )
653ad2ant1 1051 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  ( J  tX  J )  e.  Top )
76adantr 472 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =  (/) )  ->  ( J  tX  J )  e.  Top )
8 0nei 20221 . . . 4  |-  ( ( J  tX  J )  e.  Top  ->  (/)  e.  ( ( nei `  ( J  tX  J ) ) `
 (/) ) )
97, 8syl 17 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =  (/) )  ->  (/)  e.  ( ( nei `  ( J 
tX  J ) ) `
 (/) ) )
10 coeq1 4997 . . . . . . 7  |-  ( M  =  (/)  ->  ( M  o.  V )  =  ( (/)  o.  V
) )
11 co01 5357 . . . . . . 7  |-  ( (/)  o.  V )  =  (/)
1210, 11syl6eq 2521 . . . . . 6  |-  ( M  =  (/)  ->  ( M  o.  V )  =  (/) )
1312coeq2d 5002 . . . . 5  |-  ( M  =  (/)  ->  ( V  o.  ( M  o.  V ) )  =  ( V  o.  (/) ) )
14 co02 5356 . . . . 5  |-  ( V  o.  (/) )  =  (/)
1513, 14syl6eq 2521 . . . 4  |-  ( M  =  (/)  ->  ( V  o.  ( M  o.  V ) )  =  (/) )
1615adantl 473 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =  (/) )  ->  ( V  o.  ( M  o.  V
) )  =  (/) )
17 simpr 468 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =  (/) )  ->  M  =  (/) )
1817fveq2d 5883 . . 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 2564 . 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 472 . . . . . 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 1033 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  U  e.  (UnifOn `  X ) )
2221, 3syl 17 . . . . . . . . 9  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  J  e.  Top )
23 simpl2l 1083 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  V  e.  U )
24 simp3 1032 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  M  C_  ( X  X.  X ) )
2524sselda 3418 . . . . . . . . . . 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 6842 . . . . . . . . . . 11  |-  ( r  e.  ( X  X.  X )  ->  ( 1st `  r )  e.  X )
2725, 26syl 17 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  ( 1st `  r )  e.  X
)
281utopsnnei 21342 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  ( 1st `  r )  e.  X )  ->  ( V " { ( 1st `  r ) } )  e.  ( ( nei `  J ) `  {
( 1st `  r
) } ) )
2921, 23, 27, 28syl3anc 1292 . . . . . . . . 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 6843 . . . . . . . . . . 11  |-  ( r  e.  ( X  X.  X )  ->  ( 2nd `  r )  e.  X )
3125, 30syl 17 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  ( 2nd `  r )  e.  X
)
321utopsnnei 21342 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  ( 2nd `  r )  e.  X )  ->  ( V " { ( 2nd `  r ) } )  e.  ( ( nei `  J ) `  {
( 2nd `  r
) } ) )
3321, 23, 31, 32syl3anc 1292 . . . . . . . . 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 2471 . . . . . . . . . 10  |-  U. J  =  U. J
3534, 34neitx 20699 . . . . . . . . 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 1293 . . . . . . . 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 5889 . . . . . . . . . 10  |-  ( 1st `  r )  e.  _V
38 fvex 5889 . . . . . . . . . 10  |-  ( 2nd `  r )  e.  _V
3937, 38xpsn 6082 . . . . . . . . 9  |-  ( { ( 1st `  r
) }  X.  {
( 2nd `  r
) } )  =  { <. ( 1st `  r
) ,  ( 2nd `  r ) >. }
4039fveq2i 5882 . . . . . . . 8  |-  ( ( nei `  ( J 
tX  J ) ) `
 ( { ( 1st `  r ) }  X.  { ( 2nd `  r ) } ) )  =  ( ( nei `  ( J  tX  J ) ) `
 { <. ( 1st `  r ) ,  ( 2nd `  r
) >. } )
4136, 40syl6eleq 2559 . . . . . . 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 472 . . . . . . . . . . 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 4946 . . . . . . . . . . . . 13  |-  ( X  X.  X )  C_  ( _V  X.  _V )
44 sstr 3426 . . . . . . . . . . . . 13  |-  ( ( M  C_  ( X  X.  X )  /\  ( X  X.  X )  C_  ( _V  X.  _V )
)  ->  M  C_  ( _V  X.  _V ) )
4543, 44mpan2 685 . . . . . . . . . . . 12  |-  ( M 
C_  ( X  X.  X )  ->  M  C_  ( _V  X.  _V ) )
46 df-rel 4846 . . . . . . . . . . . 12  |-  ( Rel 
M  <->  M  C_  ( _V 
X.  _V ) )
4745, 46sylibr 217 . . . . . . . . . . 11  |-  ( M 
C_  ( X  X.  X )  ->  Rel  M )
4842, 47syl 17 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  Rel  M )
49 1st2nd 6858 . . . . . . . . . 10  |-  ( ( Rel  M  /\  r  e.  M )  ->  r  =  <. ( 1st `  r
) ,  ( 2nd `  r ) >. )
5048, 49sylancom 680 . . . . . . . . 9  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  r  =  <. ( 1st `  r
) ,  ( 2nd `  r ) >. )
5150sneqd 3971 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  r  e.  M
)  ->  { r }  =  { <. ( 1st `  r ) ,  ( 2nd `  r
) >. } )
5251fveq2d 5883 . . . . . . 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 2552 . . . . . 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 4947 . . . . . . . . . . 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 6858 . . . . . . . . . 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 680 . . . . . . . . 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 1070 . . . . . . . . . . . . 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 470 . . . . . . . . . . . 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 1069 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . 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 21304 . . . . . . . . . . . . . 14  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  Rel  V )
6360, 61, 62syl2anc 673 . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . 14  |-  ( z  e.  ( ( V
" { ( 1st `  r ) } )  X.  ( V " { ( 2nd `  r
) } ) )  ->  ( 1st `  z
)  e.  ( V
" { ( 1st `  r ) } ) )
6564adantl 473 . . . . . . . . . . . . 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 5198 . . . . . . . . . . . . . 14  |-  ( Rel 
V  ->  ( ( 1st `  z )  e.  ( V " {
( 1st `  r
) } )  <->  ( 1st `  r ) V ( 1st `  z ) ) )
6766biimpa 492 . . . . . . . . . . . . 13  |-  ( ( Rel  V  /\  ( 1st `  z )  e.  ( V " {
( 1st `  r
) } ) )  ->  ( 1st `  r
) V ( 1st `  z ) )
6863, 65, 67syl2anc 673 . . . . . . . . . . . 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 5889 . . . . . . . . . . . . . . 15  |-  ( 1st `  z )  e.  _V
7037, 69brcnv 5022 . . . . . . . . . . . . . 14  |-  ( ( 1st `  r ) `' V ( 1st `  z
)  <->  ( 1st `  z
) V ( 1st `  r ) )
71 breq 4397 . . . . . . . . . . . . . 14  |-  ( `' V  =  V  -> 
( ( 1st `  r
) `' V ( 1st `  z )  <-> 
( 1st `  r
) V ( 1st `  z ) ) )
7270, 71syl5bbr 267 . . . . . . . . . . . . 13  |-  ( `' V  =  V  -> 
( ( 1st `  z
) V ( 1st `  r )  <->  ( 1st `  r ) V ( 1st `  z ) ) )
7372biimpar 493 . . . . . . . . . . . 12  |-  ( ( `' V  =  V  /\  ( 1st `  r
) V ( 1st `  z ) )  -> 
( 1st `  z
) V ( 1st `  r ) )
7459, 68, 73syl2anc 673 . . . . . . . . . . 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 1071 . . . . . . . . . . . 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 770 . . . . . . . . . . . 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 6861 . . . . . . . . . . . . 13  |-  ( ( Rel  M  /\  r  e.  M )  ->  ( 1st `  r ) M ( 2nd `  r
) )
7847, 77sylan 479 . . . . . . . . . . . 12  |-  ( ( M  C_  ( X  X.  X )  /\  r  e.  M )  ->  ( 1st `  r ) M ( 2nd `  r
) )
7975, 76, 78syl2anc 673 . . . . . . . . . . 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 6843 . . . . . . . . . . . . 13  |-  ( z  e.  ( ( V
" { ( 1st `  r ) } )  X.  ( V " { ( 2nd `  r
) } ) )  ->  ( 2nd `  z
)  e.  ( V
" { ( 2nd `  r ) } ) )
8180adantl 473 . . . . . . . . . . . 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 5198 . . . . . . . . . . . . 13  |-  ( Rel 
V  ->  ( ( 2nd `  z )  e.  ( V " {
( 2nd `  r
) } )  <->  ( 2nd `  r ) V ( 2nd `  z ) ) )
8382biimpa 492 . . . . . . . . . . . 12  |-  ( ( Rel  V  /\  ( 2nd `  z )  e.  ( V " {
( 2nd `  r
) } ) )  ->  ( 2nd `  r
) V ( 2nd `  z ) )
8463, 81, 83syl2anc 673 . . . . . . . . . . 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 1208 . . . . . . . . . . . . 13  |-  ( ( 1st `  z )  e.  _V  /\  ( 2nd `  r )  e. 
_V  /\  ( 1st `  r )  e.  _V )
86 brcogw 5008 . . . . . . . . . . . . 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 684 . . . . . . . . . . . 12  |-  ( ( ( 1st `  z
) V ( 1st `  r )  /\  ( 1st `  r ) M ( 2nd `  r
) )  ->  ( 1st `  z ) ( M  o.  V ) ( 2nd `  r
) )
88 fvex 5889 . . . . . . . . . . . . . 14  |-  ( 2nd `  z )  e.  _V
8969, 88, 383pm3.2i 1208 . . . . . . . . . . . . 13  |-  ( ( 1st `  z )  e.  _V  /\  ( 2nd `  z )  e. 
_V  /\  ( 2nd `  r )  e.  _V )
90 brcogw 5008 . . . . . . . . . . . . 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 684 . . . . . . . . . . . 12  |-  ( ( ( 1st `  z
) ( M  o.  V ) ( 2nd `  r )  /\  ( 2nd `  r ) V ( 2nd `  z
) )  ->  ( 1st `  z ) ( V  o.  ( M  o.  V ) ) ( 2nd `  z
) )
9287, 91sylan 479 . . . . . . . . . . 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 1291 . . . . . . . . . 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 4396 . . . . . . . . . 10  |-  ( ( 1st `  z ) ( V  o.  ( M  o.  V )
) ( 2nd `  z
)  <->  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  e.  ( V  o.  ( M  o.  V ) ) )
9593, 94sylib 201 . . . . . . . . 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 2549 . . . . . . . 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 441 . . . . . . 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 3424 . . . . . 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 1030 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  U  e.  (UnifOn `  X ) )
100 simp2l 1056 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  V  e.  U
)
101 ustssxp 21297 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( X  X.  X
) )
10299, 100, 101syl2anc 673 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  V  C_  ( X  X.  X ) )
103 coss1 4995 . . . . . . . . . 10  |-  ( V 
C_  ( X  X.  X )  ->  ( V  o.  ( M  o.  V ) )  C_  ( ( X  X.  X )  o.  ( M  o.  V )
) )
104102, 103syl 17 . . . . . . . . 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 4995 . . . . . . . . . . . 12  |-  ( M 
C_  ( X  X.  X )  ->  ( M  o.  V )  C_  ( ( X  X.  X )  o.  V
) )
10624, 105syl 17 . . . . . . . . . . 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 4996 . . . . . . . . . . . . 13  |-  ( V 
C_  ( X  X.  X )  ->  (
( X  X.  X
)  o.  V ) 
C_  ( ( X  X.  X )  o.  ( X  X.  X
) ) )
108 xpcoid 5384 . . . . . . . . . . . . 13  |-  ( ( X  X.  X )  o.  ( X  X.  X ) )  =  ( X  X.  X
)
109107, 108syl6sseq 3464 . . . . . . . . . . . 12  |-  ( V 
C_  ( X  X.  X )  ->  (
( X  X.  X
)  o.  V ) 
C_  ( X  X.  X ) )
110102, 109syl 17 . . . . . . . . . . 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 3428 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  ( M  o.  V )  C_  ( X  X.  X ) )
112 coss2 4996 . . . . . . . . . . 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 3464 . . . . . . . . . 10  |-  ( ( M  o.  V ) 
C_  ( X  X.  X )  ->  (
( X  X.  X
)  o.  ( M  o.  V ) ) 
C_  ( X  X.  X ) )
114111, 113syl 17 . . . . . . . . 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 3428 . . . . . . . 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 21328 . . . . . . . . . . . 12  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. (unifTop `  U )
)
1171unieqi 4199 . . . . . . . . . . . 12  |-  U. J  =  U. (unifTop `  U
)
118116, 117syl6eqr 2523 . . . . . . . . . . 11  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. J )
119118sqxpeqd 4865 . . . . . . . . . 10  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  =  ( U. J  X.  U. J ) )
12034, 34txuni 20684 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  J  e.  Top )  ->  ( U. J  X.  U. J )  =  U. ( J  tX  J ) )
1213, 3, 120syl2anc 673 . . . . . . . . . 10  |-  ( U  e.  (UnifOn `  X
)  ->  ( U. J  X.  U. J )  =  U. ( J 
tX  J ) )
122119, 121eqtrd 2505 . . . . . . . . 9  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  =  U. ( J  tX  J ) )
1231223ad2ant1 1051 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  ( X  X.  X )  =  U. ( J  tX  J ) )
124115, 123sseqtrd 3454 . . . . . . 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 472 . . . . . 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 2471 . . . . . . 7  |-  U. ( J  tX  J )  = 
U. ( J  tX  J )
127126ssnei2 20209 . . . . . 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 1293 . . . . 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 2809 . . . 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 472 . . 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 472 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =/=  (/) )  -> 
( J  tX  J
)  e.  Top )
13224, 123sseqtrd 3454 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  ->  M  C_  U. ( J  tX  J ) )
133132adantr 472 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =/=  (/) )  ->  M  C_  U. ( J 
tX  J ) )
134 simpr 468 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( V  e.  U  /\  `' V  =  V
)  /\  M  C_  ( X  X.  X ) )  /\  M  =/=  (/) )  ->  M  =/=  (/) )
135126neips 20206 . . . 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 1292 . . 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 240 . 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 2730 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 setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   _Vcvv 3031    C_ wss 3390   (/)c0 3722   {csn 3959   <.cop 3965   U.cuni 4190   class class class wbr 4395    X. cxp 4837   `'ccnv 4838   "cima 4842    o. ccom 4843   Rel wrel 4844   ` cfv 5589  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811   Topctop 19994   neicnei 20190    tX ctx 20652  UnifOncust 21292  unifTopcutop 21323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-fin 7591  df-fi 7943  df-topgen 15420  df-top 19998  df-bases 19999  df-topon 20000  df-nei 20191  df-tx 20654  df-ust 21293  df-utop 21324
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator