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Theorem isfne 25680
Description: The predicate " B is finer than  A." This property is, in a sense, the opposite of refinement, as refinement requires every element to be a subset of an element of the original and fineness requires that every element of the original have a subset in the finer cover containing every point. I do not know of a literature reference for this. (Contributed by Jeff Hankins, 28-Sep-2009.)
Hypotheses
Ref Expression
isfne.1  |-  X  = 
U. A
isfne.2  |-  Y  = 
U. B
Assertion
Ref Expression
isfne  |-  ( B  e.  C  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) ) )
Distinct variable groups:    x, A    x, B    x, C
Allowed substitution hints:    X( x)    Y( x)

Proof of Theorem isfne
Dummy variables  s 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnerel 25679 . . . . 5  |-  Rel  Fne
21brrelexi 4729 . . . 4  |-  ( A Fne B  ->  A  e.  _V )
32anim1i 551 . . 3  |-  ( ( A Fne B  /\  B  e.  C )  ->  ( A  e.  _V  /\  B  e.  C ) )
43ancoms 439 . 2  |-  ( ( B  e.  C  /\  A Fne B )  -> 
( A  e.  _V  /\  B  e.  C ) )
5 simpr 447 . . . . 5  |-  ( ( B  e.  C  /\  X  =  Y )  ->  X  =  Y )
6 isfne.1 . . . . 5  |-  X  = 
U. A
7 isfne.2 . . . . 5  |-  Y  = 
U. B
85, 6, 73eqtr3g 2338 . . . 4  |-  ( ( B  e.  C  /\  X  =  Y )  ->  U. A  =  U. B )
9 simpr 447 . . . . . . 7  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  U. A  = 
U. B )
10 uniexg 4517 . . . . . . . 8  |-  ( B  e.  C  ->  U. B  e.  _V )
1110adantr 451 . . . . . . 7  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  U. B  e. 
_V )
129, 11eqeltrd 2357 . . . . . 6  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  U. A  e. 
_V )
13 uniexb 4563 . . . . . 6  |-  ( A  e.  _V  <->  U. A  e. 
_V )
1412, 13sylibr 203 . . . . 5  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  A  e.  _V )
15 simpl 443 . . . . 5  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  B  e.  C )
1614, 15jca 518 . . . 4  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  ( A  e.  _V  /\  B  e.  C ) )
178, 16syldan 456 . . 3  |-  ( ( B  e.  C  /\  X  =  Y )  ->  ( A  e.  _V  /\  B  e.  C ) )
1817adantrr 697 . 2  |-  ( ( B  e.  C  /\  ( X  =  Y  /\  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) )  ->  ( A  e.  _V  /\  B  e.  C ) )
19 unieq 3836 . . . . . 6  |-  ( r  =  A  ->  U. r  =  U. A )
2019, 6syl6eqr 2333 . . . . 5  |-  ( r  =  A  ->  U. r  =  X )
2120eqeq1d 2291 . . . 4  |-  ( r  =  A  ->  ( U. r  =  U. s 
<->  X  =  U. s
) )
22 raleq 2736 . . . 4  |-  ( r  =  A  ->  ( A. x  e.  r  x  C_  U. ( s  i^i  ~P x )  <->  A. x  e.  A  x  C_  U. ( s  i^i  ~P x ) ) )
2321, 22anbi12d 691 . . 3  |-  ( r  =  A  ->  (
( U. r  = 
U. s  /\  A. x  e.  r  x  C_ 
U. ( s  i^i 
~P x ) )  <-> 
( X  =  U. s  /\  A. x  e.  A  x  C_  U. (
s  i^i  ~P x
) ) ) )
24 unieq 3836 . . . . . 6  |-  ( s  =  B  ->  U. s  =  U. B )
2524, 7syl6eqr 2333 . . . . 5  |-  ( s  =  B  ->  U. s  =  Y )
2625eqeq2d 2294 . . . 4  |-  ( s  =  B  ->  ( X  =  U. s  <->  X  =  Y ) )
27 ineq1 3363 . . . . . . 7  |-  ( s  =  B  ->  (
s  i^i  ~P x
)  =  ( B  i^i  ~P x ) )
2827unieqd 3838 . . . . . 6  |-  ( s  =  B  ->  U. (
s  i^i  ~P x
)  =  U. ( B  i^i  ~P x ) )
2928sseq2d 3206 . . . . 5  |-  ( s  =  B  ->  (
x  C_  U. (
s  i^i  ~P x
)  <->  x  C_  U. ( B  i^i  ~P x ) ) )
3029ralbidv 2563 . . . 4  |-  ( s  =  B  ->  ( A. x  e.  A  x  C_  U. ( s  i^i  ~P x )  <->  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) )
3126, 30anbi12d 691 . . 3  |-  ( s  =  B  ->  (
( X  =  U. s  /\  A. x  e.  A  x  C_  U. (
s  i^i  ~P x
) )  <->  ( X  =  Y  /\  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) ) )
32 df-fne 25675 . . 3  |-  Fne  =  { <. r ,  s
>.  |  ( U. r  =  U. s  /\  A. x  e.  r  x  C_  U. (
s  i^i  ~P x
) ) }
3323, 31, 32brabg 4284 . 2  |-  ( ( A  e.  _V  /\  B  e.  C )  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  x 
C_  U. ( B  i^i  ~P x ) ) ) )
344, 18, 33pm5.21nd 868 1  |-  ( B  e.  C  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   class class class wbr 4023   Fnecfne 25671
This theorem is referenced by:  isfne4  25681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-fne 25675
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