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Theorem isfne4 26372
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 26370 . . 3  |-  Rel  Fne
21brrelex2i 4746 . 2  |-  ( A Fne B  ->  B  e.  _V )
3 simpl 443 . . . . 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 2351 . . . 4  |-  ( ( X  =  Y  /\  A  C_  ( topGen `  B
) )  ->  U. A  =  U. B )
7 fvex 5555 . . . . . . 7  |-  ( topGen `  B )  e.  _V
87ssex 4174 . . . . . 6  |-  ( A 
C_  ( topGen `  B
)  ->  A  e.  _V )
98adantl 452 . . . . 5  |-  ( ( X  =  Y  /\  A  C_  ( topGen `  B
) )  ->  A  e.  _V )
10 uniexb 4579 . . . . 5  |-  ( A  e.  _V  <->  U. A  e. 
_V )
119, 10sylib 188 . . . 4  |-  ( ( X  =  Y  /\  A  C_  ( topGen `  B
) )  ->  U. A  e.  _V )
126, 11eqeltrrd 2371 . . 3  |-  ( ( X  =  Y  /\  A  C_  ( topGen `  B
) )  ->  U. B  e.  _V )
13 uniexb 4579 . . 3  |-  ( B  e.  _V  <->  U. B  e. 
_V )
1412, 13sylibr 203 . 2  |-  ( ( X  =  Y  /\  A  C_  ( topGen `  B
) )  ->  B  e.  _V )
154, 5isfne 26371 . . 3  |-  ( B  e.  _V  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) ) )
16 dfss3 3183 . . . . 5  |-  ( A 
C_  ( topGen `  B
)  <->  A. x  e.  A  x  e.  ( topGen `  B ) )
17 eltg 16711 . . . . . 6  |-  ( B  e.  _V  ->  (
x  e.  ( topGen `  B )  <->  x  C_  U. ( B  i^i  ~P x ) ) )
1817ralbidv 2576 . . . . 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 248 . . . 4  |-  ( B  e.  _V  ->  ( A  C_  ( topGen `  B
)  <->  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) )
2019anbi2d 684 . . 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 247 . 2  |-  ( B  e.  _V  ->  ( A Fne B  <->  ( X  =  Y  /\  A  C_  ( topGen `  B )
) ) )
222, 14, 21pm5.21nii 342 1  |-  ( A Fne B  <->  ( X  =  Y  /\  A  C_  ( topGen `  B )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   class class class wbr 4039   ` cfv 5271   topGenctg 13358   Fnecfne 26362
This theorem is referenced by:  isfne4b  26373  isfne2  26374  isfne3  26375  fnebas  26376  fnetg  26377  topfne  26393  fnemeet1  26418  fnemeet2  26419  fnejoin1  26420  fnejoin2  26421
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-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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