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Theorem fneval 26287
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 4029 . . 3  |-  ( A  .~  B  <->  A ( Fne  i^i  `' Fne ) B )
3 brin 4070 . . . 4  |-  ( A ( Fne  i^i  `' Fne ) B  <->  ( A Fne B  /\  A `' Fne B ) )
4 fnerel 26267 . . . . . 6  |-  Rel  Fne
54relbrcnv 5054 . . . . 5  |-  ( A `' Fne B  <->  B Fne A )
65anbi2i 675 . . . 4  |-  ( ( A Fne B  /\  A `' Fne B )  <->  ( A Fne B  /\  B Fne A ) )
73, 6bitri 240 . . 3  |-  ( A ( Fne  i^i  `' Fne ) B  <->  ( A Fne B  /\  B Fne A ) )
82, 7bitri 240 . 2  |-  ( A  .~  B  <->  ( A Fne B  /\  B Fne A ) )
9 eqid 2283 . . . . . 6  |-  U. A  =  U. A
10 eqid 2283 . . . . . 6  |-  U. B  =  U. B
119, 10isfne4b 26270 . . . . 5  |-  ( B  e.  W  ->  ( A Fne B  <->  ( U. A  =  U. B  /\  ( topGen `  A )  C_  ( topGen `  B )
) ) )
1210, 9isfne4b 26270 . . . . . 6  |-  ( A  e.  V  ->  ( B Fne A  <->  ( U. B  =  U. A  /\  ( topGen `  B )  C_  ( topGen `  A )
) ) )
13 eqcom 2285 . . . . . . 7  |-  ( U. B  =  U. A  <->  U. A  = 
U. B )
1413anbi1i 676 . . . . . 6  |-  ( ( U. B  =  U. A  /\  ( topGen `  B
)  C_  ( topGen `  A ) )  <->  ( U. A  =  U. B  /\  ( topGen `  B )  C_  ( topGen `  A )
) )
1512, 14syl6bb 252 . . . . 5  |-  ( A  e.  V  ->  ( B Fne A  <->  ( U. A  =  U. B  /\  ( topGen `  B )  C_  ( topGen `  A )
) ) )
1611, 15bi2anan9r 844 . . . 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 3194 . . . . . 6  |-  ( (
topGen `  A )  =  ( topGen `  B )  <->  ( ( topGen `  A )  C_  ( topGen `  B )  /\  ( topGen `  B )  C_  ( topGen `  A )
) )
1817anbi2i 675 . . . . 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 801 . . . . 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 240 . . . 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 254 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A Fne B  /\  B Fne A
)  <->  ( U. A  =  U. B  /\  ( topGen `
 A )  =  ( topGen `  B )
) ) )
22 unieq 3836 . . . . 5  |-  ( (
topGen `  A )  =  ( topGen `  B )  ->  U. ( topGen `  A
)  =  U. ( topGen `
 B ) )
23 unitg 16705 . . . . . 6  |-  ( A  e.  V  ->  U. ( topGen `
 A )  = 
U. A )
24 unitg 16705 . . . . . 6  |-  ( B  e.  W  ->  U. ( topGen `
 B )  = 
U. B )
2523, 24eqeqan12d 2298 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( U. ( topGen `  A )  =  U. ( topGen `  B )  <->  U. A  =  U. B
) )
2622, 25syl5ib 210 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( topGen `  A
)  =  ( topGen `  B )  ->  U. A  =  U. B ) )
2726pm4.71rd 616 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( topGen `  A
)  =  ( topGen `  B )  <->  ( U. A  =  U. B  /\  ( topGen `  A )  =  ( topGen `  B
) ) ) )
2821, 27bitr4d 247 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A Fne B  /\  B Fne A
)  <->  ( topGen `  A
)  =  ( topGen `  B ) ) )
298, 28syl5bb 248 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  .~  B  <->  (
topGen `  A )  =  ( topGen `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   U.cuni 3827   class class class wbr 4023   `'ccnv 4688   ` cfv 5255   topGenctg 13342   Fnecfne 26259
This theorem is referenced by:  fneer  26288  topfneec  26291  topfneec2  26292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-topgen 13344  df-fne 26263
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