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Theorem isfne4 26247
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 26245 . . 3  |-  Rel  Fne
21brrelex2i 4886 . 2  |-  ( A Fne B  ->  B  e.  _V )
3 simpl 444 . . . . 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 2467 . . . 4  |-  ( ( X  =  Y  /\  A  C_  ( topGen `  B
) )  ->  U. A  =  U. B )
7 fvex 5709 . . . . . . 7  |-  ( topGen `  B )  e.  _V
87ssex 4315 . . . . . 6  |-  ( A 
C_  ( topGen `  B
)  ->  A  e.  _V )
98adantl 453 . . . . 5  |-  ( ( X  =  Y  /\  A  C_  ( topGen `  B
) )  ->  A  e.  _V )
10 uniexb 4719 . . . . 5  |-  ( A  e.  _V  <->  U. A  e. 
_V )
119, 10sylib 189 . . . 4  |-  ( ( X  =  Y  /\  A  C_  ( topGen `  B
) )  ->  U. A  e.  _V )
126, 11eqeltrrd 2487 . . 3  |-  ( ( X  =  Y  /\  A  C_  ( topGen `  B
) )  ->  U. B  e.  _V )
13 uniexb 4719 . . 3  |-  ( B  e.  _V  <->  U. B  e. 
_V )
1412, 13sylibr 204 . 2  |-  ( ( X  =  Y  /\  A  C_  ( topGen `  B
) )  ->  B  e.  _V )
154, 5isfne 26246 . . 3  |-  ( B  e.  _V  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) ) )
16 dfss3 3306 . . . . 5  |-  ( A 
C_  ( topGen `  B
)  <->  A. x  e.  A  x  e.  ( topGen `  B ) )
17 eltg 16985 . . . . . 6  |-  ( B  e.  _V  ->  (
x  e.  ( topGen `  B )  <->  x  C_  U. ( B  i^i  ~P x ) ) )
1817ralbidv 2694 . . . . 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 249 . . . 4  |-  ( B  e.  _V  ->  ( A  C_  ( topGen `  B
)  <->  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) )
2019anbi2d 685 . . 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 248 . 2  |-  ( B  e.  _V  ->  ( A Fne B  <->  ( X  =  Y  /\  A  C_  ( topGen `  B )
) ) )
222, 14, 21pm5.21nii 343 1  |-  ( A Fne B  <->  ( X  =  Y  /\  A  C_  ( topGen `  B )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   _Vcvv 2924    i^i cin 3287    C_ wss 3288   ~Pcpw 3767   U.cuni 3983   class class class wbr 4180   ` cfv 5421   topGenctg 13628   Fnecfne 26237
This theorem is referenced by:  isfne4b  26248  isfne2  26249  isfne3  26250  fnebas  26251  fnetg  26252  topfne  26268  fnemeet1  26293  fnemeet2  26294  fnejoin1  26295  fnejoin2  26296
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-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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