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Theorem isref 26279
Description: The property of being a refinement of a cover. Dr. Nyikos once commented in class that the term "refinement" is actually misleading and that people are inclined to confuse it with the notion defined in isfne 26268. On the other hand, the two concepts do seem to have a dual relationship. (Contributed by Jeff Hankins, 18-Jan-2010.)
Hypotheses
Ref Expression
isref.1  |-  X  = 
U. A
isref.2  |-  Y  = 
U. B
Assertion
Ref Expression
isref  |-  ( B  e.  C  ->  ( A Ref B  <->  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y
) ) )
Distinct variable groups:    x, y, A    x, B
Allowed substitution hints:    B( y)    C( x, y)    X( x, y)    Y( x, y)

Proof of Theorem isref
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 refrel 26278 . . . . 5  |-  Rel  Ref
21brrelexi 4729 . . . 4  |-  ( A Ref B  ->  A  e.  _V )
32adantl 452 . . 3  |-  ( ( B  e.  C  /\  A Ref B )  ->  A  e.  _V )
4 simpl 443 . . 3  |-  ( ( B  e.  C  /\  A Ref B )  ->  B  e.  C )
53, 4jca 518 . 2  |-  ( ( B  e.  C  /\  A Ref B )  -> 
( A  e.  _V  /\  B  e.  C ) )
6 simpr 447 . . . . . . 7  |-  ( ( B  e.  C  /\  X  =  Y )  ->  X  =  Y )
7 isref.1 . . . . . . 7  |-  X  = 
U. A
8 isref.2 . . . . . . 7  |-  Y  = 
U. B
96, 7, 83eqtr3g 2338 . . . . . 6  |-  ( ( B  e.  C  /\  X  =  Y )  ->  U. A  =  U. B )
10 uniexg 4517 . . . . . . 7  |-  ( B  e.  C  ->  U. B  e.  _V )
1110adantr 451 . . . . . 6  |-  ( ( B  e.  C  /\  X  =  Y )  ->  U. B  e.  _V )
129, 11eqeltrd 2357 . . . . 5  |-  ( ( B  e.  C  /\  X  =  Y )  ->  U. A  e.  _V )
13 uniexb 4563 . . . . 5  |-  ( A  e.  _V  <->  U. A  e. 
_V )
1412, 13sylibr 203 . . . 4  |-  ( ( B  e.  C  /\  X  =  Y )  ->  A  e.  _V )
1514adantrr 697 . . 3  |-  ( ( B  e.  C  /\  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y ) )  ->  A  e.  _V )
16 simpl 443 . . 3  |-  ( ( B  e.  C  /\  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y ) )  ->  B  e.  C
)
1715, 16jca 518 . 2  |-  ( ( B  e.  C  /\  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y ) )  ->  ( A  e. 
_V  /\  B  e.  C ) )
18 unieq 3836 . . . . . 6  |-  ( a  =  A  ->  U. a  =  U. A )
1918, 7syl6eqr 2333 . . . . 5  |-  ( a  =  A  ->  U. a  =  X )
2019eqeq1d 2291 . . . 4  |-  ( a  =  A  ->  ( U. a  =  U. b 
<->  X  =  U. b
) )
21 rexeq 2737 . . . . 5  |-  ( a  =  A  ->  ( E. y  e.  a  x  C_  y  <->  E. y  e.  A  x  C_  y
) )
2221ralbidv 2563 . . . 4  |-  ( a  =  A  ->  ( A. x  e.  b  E. y  e.  a  x  C_  y  <->  A. x  e.  b  E. y  e.  A  x  C_  y
) )
2320, 22anbi12d 691 . . 3  |-  ( a  =  A  ->  (
( U. a  = 
U. b  /\  A. x  e.  b  E. y  e.  a  x  C_  y )  <->  ( X  =  U. b  /\  A. x  e.  b  E. y  e.  A  x  C_  y ) ) )
24 unieq 3836 . . . . . 6  |-  ( b  =  B  ->  U. b  =  U. B )
2524, 8syl6eqr 2333 . . . . 5  |-  ( b  =  B  ->  U. b  =  Y )
2625eqeq2d 2294 . . . 4  |-  ( b  =  B  ->  ( X  =  U. b  <->  X  =  Y ) )
27 raleq 2736 . . . 4  |-  ( b  =  B  ->  ( A. x  e.  b  E. y  e.  A  x  C_  y  <->  A. x  e.  B  E. y  e.  A  x  C_  y
) )
2826, 27anbi12d 691 . . 3  |-  ( b  =  B  ->  (
( X  =  U. b  /\  A. x  e.  b  E. y  e.  A  x  C_  y
)  <->  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y
) ) )
29 df-ref 26264 . . 3  |-  Ref  =  { <. a ,  b
>.  |  ( U. a  =  U. b  /\  A. x  e.  b  E. y  e.  a  x  C_  y ) }
3023, 28, 29brabg 4284 . 2  |-  ( ( A  e.  _V  /\  B  e.  C )  ->  ( A Ref B  <->  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y ) ) )
315, 17, 30pm5.21nd 868 1  |-  ( B  e.  C  ->  ( A Ref B  <->  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   U.cuni 3827   class class class wbr 4023   Refcref 26260
This theorem is referenced by:  refbas  26280  refssex  26281  ssref  26283  refref  26285  reftr  26289  fnessref  26293  refssfne  26294
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-ref 26264
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