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Theorem fneval 26381
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 4221 . . 3  |-  ( A  .~  B  <->  A ( Fne  i^i  `' Fne ) B )
3 brin 4262 . . . 4  |-  ( A ( Fne  i^i  `' Fne ) B  <->  ( A Fne B  /\  A `' Fne B ) )
4 fnerel 26361 . . . . . 6  |-  Rel  Fne
54relbrcnv 5248 . . . . 5  |-  ( A `' Fne B  <->  B Fne A )
65anbi2i 677 . . . 4  |-  ( ( A Fne B  /\  A `' Fne B )  <->  ( A Fne B  /\  B Fne A ) )
73, 6bitri 242 . . 3  |-  ( A ( Fne  i^i  `' Fne ) B  <->  ( A Fne B  /\  B Fne A ) )
82, 7bitri 242 . 2  |-  ( A  .~  B  <->  ( A Fne B  /\  B Fne A ) )
9 eqid 2438 . . . . . 6  |-  U. A  =  U. A
10 eqid 2438 . . . . . 6  |-  U. B  =  U. B
119, 10isfne4b 26364 . . . . 5  |-  ( B  e.  W  ->  ( A Fne B  <->  ( U. A  =  U. B  /\  ( topGen `  A )  C_  ( topGen `  B )
) ) )
1210, 9isfne4b 26364 . . . . . 6  |-  ( A  e.  V  ->  ( B Fne A  <->  ( U. B  =  U. A  /\  ( topGen `  B )  C_  ( topGen `  A )
) ) )
13 eqcom 2440 . . . . . . 7  |-  ( U. B  =  U. A  <->  U. A  = 
U. B )
1413anbi1i 678 . . . . . 6  |-  ( ( U. B  =  U. A  /\  ( topGen `  B
)  C_  ( topGen `  A ) )  <->  ( U. A  =  U. B  /\  ( topGen `  B )  C_  ( topGen `  A )
) )
1512, 14syl6bb 254 . . . . 5  |-  ( A  e.  V  ->  ( B Fne A  <->  ( U. A  =  U. B  /\  ( topGen `  B )  C_  ( topGen `  A )
) ) )
1611, 15bi2anan9r 846 . . . 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 3365 . . . . . 6  |-  ( (
topGen `  A )  =  ( topGen `  B )  <->  ( ( topGen `  A )  C_  ( topGen `  B )  /\  ( topGen `  B )  C_  ( topGen `  A )
) )
1817anbi2i 677 . . . . 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 803 . . . . 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 242 . . . 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 256 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A Fne B  /\  B Fne A
)  <->  ( U. A  =  U. B  /\  ( topGen `
 A )  =  ( topGen `  B )
) ) )
22 unieq 4026 . . . . 5  |-  ( (
topGen `  A )  =  ( topGen `  B )  ->  U. ( topGen `  A
)  =  U. ( topGen `
 B ) )
23 unitg 17037 . . . . . 6  |-  ( A  e.  V  ->  U. ( topGen `
 A )  = 
U. A )
24 unitg 17037 . . . . . 6  |-  ( B  e.  W  ->  U. ( topGen `
 B )  = 
U. B )
2523, 24eqeqan12d 2453 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( U. ( topGen `  A )  =  U. ( topGen `  B )  <->  U. A  =  U. B
) )
2622, 25syl5ib 212 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( topGen `  A
)  =  ( topGen `  B )  ->  U. A  =  U. B ) )
2726pm4.71rd 618 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( topGen `  A
)  =  ( topGen `  B )  <->  ( U. A  =  U. B  /\  ( topGen `  A )  =  ( topGen `  B
) ) ) )
2821, 27bitr4d 249 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A Fne B  /\  B Fne A
)  <->  ( topGen `  A
)  =  ( topGen `  B ) ) )
298, 28syl5bb 250 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  .~  B  <->  (
topGen `  A )  =  ( topGen `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3321    C_ wss 3322   U.cuni 4017   class class class wbr 4215   `'ccnv 4880   ` cfv 5457   topGenctg 13670   Fnecfne 26353
This theorem is referenced by:  fneer  26382  topfneec  26385  topfneec2  26386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-topgen 13672  df-fne 26357
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