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Theorem fneval 26265
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 4186 . . 3  |-  ( A  .~  B  <->  A ( Fne  i^i  `' Fne ) B )
3 brin 4227 . . . 4  |-  ( A ( Fne  i^i  `' Fne ) B  <->  ( A Fne B  /\  A `' Fne B ) )
4 fnerel 26245 . . . . . 6  |-  Rel  Fne
54relbrcnv 5212 . . . . 5  |-  ( A `' Fne B  <->  B Fne A )
65anbi2i 676 . . . 4  |-  ( ( A Fne B  /\  A `' Fne B )  <->  ( A Fne B  /\  B Fne A ) )
73, 6bitri 241 . . 3  |-  ( A ( Fne  i^i  `' Fne ) B  <->  ( A Fne B  /\  B Fne A ) )
82, 7bitri 241 . 2  |-  ( A  .~  B  <->  ( A Fne B  /\  B Fne A ) )
9 eqid 2412 . . . . . 6  |-  U. A  =  U. A
10 eqid 2412 . . . . . 6  |-  U. B  =  U. B
119, 10isfne4b 26248 . . . . 5  |-  ( B  e.  W  ->  ( A Fne B  <->  ( U. A  =  U. B  /\  ( topGen `  A )  C_  ( topGen `  B )
) ) )
1210, 9isfne4b 26248 . . . . . 6  |-  ( A  e.  V  ->  ( B Fne A  <->  ( U. B  =  U. A  /\  ( topGen `  B )  C_  ( topGen `  A )
) ) )
13 eqcom 2414 . . . . . . 7  |-  ( U. B  =  U. A  <->  U. A  = 
U. B )
1413anbi1i 677 . . . . . 6  |-  ( ( U. B  =  U. A  /\  ( topGen `  B
)  C_  ( topGen `  A ) )  <->  ( U. A  =  U. B  /\  ( topGen `  B )  C_  ( topGen `  A )
) )
1512, 14syl6bb 253 . . . . 5  |-  ( A  e.  V  ->  ( B Fne A  <->  ( U. A  =  U. B  /\  ( topGen `  B )  C_  ( topGen `  A )
) ) )
1611, 15bi2anan9r 845 . . . 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 3331 . . . . . 6  |-  ( (
topGen `  A )  =  ( topGen `  B )  <->  ( ( topGen `  A )  C_  ( topGen `  B )  /\  ( topGen `  B )  C_  ( topGen `  A )
) )
1817anbi2i 676 . . . . 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 802 . . . . 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 241 . . . 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 255 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A Fne B  /\  B Fne A
)  <->  ( U. A  =  U. B  /\  ( topGen `
 A )  =  ( topGen `  B )
) ) )
22 unieq 3992 . . . . 5  |-  ( (
topGen `  A )  =  ( topGen `  B )  ->  U. ( topGen `  A
)  =  U. ( topGen `
 B ) )
23 unitg 16995 . . . . . 6  |-  ( A  e.  V  ->  U. ( topGen `
 A )  = 
U. A )
24 unitg 16995 . . . . . 6  |-  ( B  e.  W  ->  U. ( topGen `
 B )  = 
U. B )
2523, 24eqeqan12d 2427 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( U. ( topGen `  A )  =  U. ( topGen `  B )  <->  U. A  =  U. B
) )
2622, 25syl5ib 211 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( topGen `  A
)  =  ( topGen `  B )  ->  U. A  =  U. B ) )
2726pm4.71rd 617 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( topGen `  A
)  =  ( topGen `  B )  <->  ( U. A  =  U. B  /\  ( topGen `  A )  =  ( topGen `  B
) ) ) )
2821, 27bitr4d 248 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A Fne B  /\  B Fne A
)  <->  ( topGen `  A
)  =  ( topGen `  B ) ) )
298, 28syl5bb 249 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  .~  B  <->  (
topGen `  A )  =  ( topGen `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3287    C_ wss 3288   U.cuni 3983   class class class wbr 4180   `'ccnv 4844   ` cfv 5421   topGenctg 13628   Fnecfne 26237
This theorem is referenced by:  fneer  26266  topfneec  26269  topfneec2  26270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-topgen 13630  df-fne 26241
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