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Theorem isfne4 26363
Description: The predicate " B is finer than  A " in terms of the topology generation function. (Contributed by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
isfne.1  |-  X  = 
U. A
isfne.2  |-  Y  = 
U. B
Assertion
Ref Expression
isfne4  |-  ( A Fne B  <->  ( X  =  Y  /\  A  C_  ( topGen `  B )
) )

Proof of Theorem isfne4
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fnerel 26361 . . 3  |-  Rel  Fne
21brrelex2i 4922 . 2  |-  ( A Fne B  ->  B  e.  _V )
3 simpl 445 . . . . 5  |-  ( ( X  =  Y  /\  A  C_  ( topGen `  B
) )  ->  X  =  Y )
4 isfne.1 . . . . 5  |-  X  = 
U. A
5 isfne.2 . . . . 5  |-  Y  = 
U. B
63, 4, 53eqtr3g 2493 . . . 4  |-  ( ( X  =  Y  /\  A  C_  ( topGen `  B
) )  ->  U. A  =  U. B )
7 fvex 5745 . . . . . . 7  |-  ( topGen `  B )  e.  _V
87ssex 4350 . . . . . 6  |-  ( A 
C_  ( topGen `  B
)  ->  A  e.  _V )
98adantl 454 . . . . 5  |-  ( ( X  =  Y  /\  A  C_  ( topGen `  B
) )  ->  A  e.  _V )
10 uniexb 4755 . . . . 5  |-  ( A  e.  _V  <->  U. A  e. 
_V )
119, 10sylib 190 . . . 4  |-  ( ( X  =  Y  /\  A  C_  ( topGen `  B
) )  ->  U. A  e.  _V )
126, 11eqeltrrd 2513 . . 3  |-  ( ( X  =  Y  /\  A  C_  ( topGen `  B
) )  ->  U. B  e.  _V )
13 uniexb 4755 . . 3  |-  ( B  e.  _V  <->  U. B  e. 
_V )
1412, 13sylibr 205 . 2  |-  ( ( X  =  Y  /\  A  C_  ( topGen `  B
) )  ->  B  e.  _V )
154, 5isfne 26362 . . 3  |-  ( B  e.  _V  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) ) )
16 dfss3 3340 . . . . 5  |-  ( A 
C_  ( topGen `  B
)  <->  A. x  e.  A  x  e.  ( topGen `  B ) )
17 eltg 17027 . . . . . 6  |-  ( B  e.  _V  ->  (
x  e.  ( topGen `  B )  <->  x  C_  U. ( B  i^i  ~P x ) ) )
1817ralbidv 2727 . . . . 5  |-  ( B  e.  _V  ->  ( A. x  e.  A  x  e.  ( topGen `  B )  <->  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) )
1916, 18syl5bb 250 . . . 4  |-  ( B  e.  _V  ->  ( A  C_  ( topGen `  B
)  <->  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) )
2019anbi2d 686 . . 3  |-  ( B  e.  _V  ->  (
( X  =  Y  /\  A  C_  ( topGen `
 B ) )  <-> 
( X  =  Y  /\  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) ) )
2115, 20bitr4d 249 . 2  |-  ( B  e.  _V  ->  ( A Fne B  <->  ( X  =  Y  /\  A  C_  ( topGen `  B )
) ) )
222, 14, 21pm5.21nii 344 1  |-  ( A Fne B  <->  ( X  =  Y  /\  A  C_  ( topGen `  B )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   class class class wbr 4215   ` cfv 5457   topGenctg 13670   Fnecfne 26353
This theorem is referenced by:  isfne4b  26364  isfne2  26365  isfne3  26366  fnebas  26367  fnetg  26368  topfne  26384  fnemeet1  26409  fnemeet2  26410  fnejoin1  26411  fnejoin2  26412
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-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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