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Theorem fneval 28708
Description: Two covers are finer than each other iff they are both bases for the same topology. (Contributed by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
fneval.1  |-  .~  =  ( Fne  i^i  `' Fne )
Assertion
Ref Expression
fneval  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  .~  B  <->  (
topGen `  A )  =  ( topGen `  B )
) )

Proof of Theorem fneval
StepHypRef Expression
1 fneval.1 . . . 4  |-  .~  =  ( Fne  i^i  `' Fne )
21breqi 4407 . . 3  |-  ( A  .~  B  <->  A ( Fne  i^i  `' Fne ) B )
3 brin 4450 . . . 4  |-  ( A ( Fne  i^i  `' Fne ) B  <->  ( A Fne B  /\  A `' Fne B ) )
4 fnerel 28688 . . . . . 6  |-  Rel  Fne
54relbrcnv 5318 . . . . 5  |-  ( A `' Fne B  <->  B Fne A )
65anbi2i 694 . . . 4  |-  ( ( A Fne B  /\  A `' Fne B )  <->  ( A Fne B  /\  B Fne A ) )
73, 6bitri 249 . . 3  |-  ( A ( Fne  i^i  `' Fne ) B  <->  ( A Fne B  /\  B Fne A ) )
82, 7bitri 249 . 2  |-  ( A  .~  B  <->  ( A Fne B  /\  B Fne A ) )
9 eqid 2454 . . . . . 6  |-  U. A  =  U. A
10 eqid 2454 . . . . . 6  |-  U. B  =  U. B
119, 10isfne4b 28691 . . . . 5  |-  ( B  e.  W  ->  ( A Fne B  <->  ( U. A  =  U. B  /\  ( topGen `  A )  C_  ( topGen `  B )
) ) )
1210, 9isfne4b 28691 . . . . . 6  |-  ( A  e.  V  ->  ( B Fne A  <->  ( U. B  =  U. A  /\  ( topGen `  B )  C_  ( topGen `  A )
) ) )
13 eqcom 2463 . . . . . . 7  |-  ( U. B  =  U. A  <->  U. A  = 
U. B )
1413anbi1i 695 . . . . . 6  |-  ( ( U. B  =  U. A  /\  ( topGen `  B
)  C_  ( topGen `  A ) )  <->  ( U. A  =  U. B  /\  ( topGen `  B )  C_  ( topGen `  A )
) )
1512, 14syl6bb 261 . . . . 5  |-  ( A  e.  V  ->  ( B Fne A  <->  ( U. A  =  U. B  /\  ( topGen `  B )  C_  ( topGen `  A )
) ) )
1611, 15bi2anan9r 869 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A Fne B  /\  B Fne A
)  <->  ( ( U. A  =  U. B  /\  ( topGen `  A )  C_  ( topGen `  B )
)  /\  ( U. A  =  U. B  /\  ( topGen `  B )  C_  ( topGen `  A )
) ) ) )
17 eqss 3480 . . . . . 6  |-  ( (
topGen `  A )  =  ( topGen `  B )  <->  ( ( topGen `  A )  C_  ( topGen `  B )  /\  ( topGen `  B )  C_  ( topGen `  A )
) )
1817anbi2i 694 . . . . 5  |-  ( ( U. A  =  U. B  /\  ( topGen `  A
)  =  ( topGen `  B ) )  <->  ( U. A  =  U. B  /\  ( ( topGen `  A
)  C_  ( topGen `  B )  /\  ( topGen `
 B )  C_  ( topGen `  A )
) ) )
19 anandi 824 . . . . 5  |-  ( ( U. A  =  U. B  /\  ( ( topGen `  A )  C_  ( topGen `
 B )  /\  ( topGen `  B )  C_  ( topGen `  A )
) )  <->  ( ( U. A  =  U. B  /\  ( topGen `  A
)  C_  ( topGen `  B ) )  /\  ( U. A  =  U. B  /\  ( topGen `  B
)  C_  ( topGen `  A ) ) ) )
2018, 19bitri 249 . . . 4  |-  ( ( U. A  =  U. B  /\  ( topGen `  A
)  =  ( topGen `  B ) )  <->  ( ( U. A  =  U. B  /\  ( topGen `  A
)  C_  ( topGen `  B ) )  /\  ( U. A  =  U. B  /\  ( topGen `  B
)  C_  ( topGen `  A ) ) ) )
2116, 20syl6bbr 263 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A Fne B  /\  B Fne A
)  <->  ( U. A  =  U. B  /\  ( topGen `
 A )  =  ( topGen `  B )
) ) )
22 unieq 4208 . . . . 5  |-  ( (
topGen `  A )  =  ( topGen `  B )  ->  U. ( topGen `  A
)  =  U. ( topGen `
 B ) )
23 unitg 18705 . . . . . 6  |-  ( A  e.  V  ->  U. ( topGen `
 A )  = 
U. A )
24 unitg 18705 . . . . . 6  |-  ( B  e.  W  ->  U. ( topGen `
 B )  = 
U. B )
2523, 24eqeqan12d 2477 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( U. ( topGen `  A )  =  U. ( topGen `  B )  <->  U. A  =  U. B
) )
2622, 25syl5ib 219 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( topGen `  A
)  =  ( topGen `  B )  ->  U. A  =  U. B ) )
2726pm4.71rd 635 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( topGen `  A
)  =  ( topGen `  B )  <->  ( U. A  =  U. B  /\  ( topGen `  A )  =  ( topGen `  B
) ) ) )
2821, 27bitr4d 256 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A Fne B  /\  B Fne A
)  <->  ( topGen `  A
)  =  ( topGen `  B ) ) )
298, 28syl5bb 257 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  .~  B  <->  (
topGen `  A )  =  ( topGen `  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    i^i cin 3436    C_ wss 3437   U.cuni 4200   class class class wbr 4401   `'ccnv 4948   ` cfv 5527   topGenctg 14496   Fnecfne 28680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535  df-topgen 14502  df-fne 28684
This theorem is referenced by:  fneer  28709  topfneec  28712  topfneec2  28713
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