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Theorem fneval 26390
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 4045 . . 3  |-  ( A  .~  B  <->  A ( Fne  i^i  `' Fne ) B )
3 brin 4086 . . . 4  |-  ( A ( Fne  i^i  `' Fne ) B  <->  ( A Fne B  /\  A `' Fne B ) )
4 fnerel 26370 . . . . . 6  |-  Rel  Fne
54relbrcnv 5070 . . . . 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 2296 . . . . . 6  |-  U. A  =  U. A
10 eqid 2296 . . . . . 6  |-  U. B  =  U. B
119, 10isfne4b 26373 . . . . 5  |-  ( B  e.  W  ->  ( A Fne B  <->  ( U. A  =  U. B  /\  ( topGen `  A )  C_  ( topGen `  B )
) ) )
1210, 9isfne4b 26373 . . . . . 6  |-  ( A  e.  V  ->  ( B Fne A  <->  ( U. B  =  U. A  /\  ( topGen `  B )  C_  ( topGen `  A )
) ) )
13 eqcom 2298 . . . . . . 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 3207 . . . . . 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 3852 . . . . 5  |-  ( (
topGen `  A )  =  ( topGen `  B )  ->  U. ( topGen `  A
)  =  U. ( topGen `
 B ) )
23 unitg 16721 . . . . . 6  |-  ( A  e.  V  ->  U. ( topGen `
 A )  = 
U. A )
24 unitg 16721 . . . . . 6  |-  ( B  e.  W  ->  U. ( topGen `
 B )  = 
U. B )
2523, 24eqeqan12d 2311 . . . . 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 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   U.cuni 3843   class class class wbr 4039   `'ccnv 4704   ` cfv 5271   topGenctg 13358   Fnecfne 26362
This theorem is referenced by:  fneer  26391  topfneec  26394  topfneec2  26395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-topgen 13360  df-fne 26366
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