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Theorem isref 26382
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 26371. 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 26381 . . . . 5  |-  Rel  Ref
21brrelexi 4745 . . . 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 2351 . . . . . 6  |-  ( ( B  e.  C  /\  X  =  Y )  ->  U. A  =  U. B )
10 uniexg 4533 . . . . . . 7  |-  ( B  e.  C  ->  U. B  e.  _V )
1110adantr 451 . . . . . 6  |-  ( ( B  e.  C  /\  X  =  Y )  ->  U. B  e.  _V )
129, 11eqeltrd 2370 . . . . 5  |-  ( ( B  e.  C  /\  X  =  Y )  ->  U. A  e.  _V )
13 uniexb 4579 . . . . 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 3852 . . . . . 6  |-  ( a  =  A  ->  U. a  =  U. A )
1918, 7syl6eqr 2346 . . . . 5  |-  ( a  =  A  ->  U. a  =  X )
2019eqeq1d 2304 . . . 4  |-  ( a  =  A  ->  ( U. a  =  U. b 
<->  X  =  U. b
) )
21 rexeq 2750 . . . . 5  |-  ( a  =  A  ->  ( E. y  e.  a  x  C_  y  <->  E. y  e.  A  x  C_  y
) )
2221ralbidv 2576 . . . 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 3852 . . . . . 6  |-  ( b  =  B  ->  U. b  =  U. B )
2524, 8syl6eqr 2346 . . . . 5  |-  ( b  =  B  ->  U. b  =  Y )
2625eqeq2d 2307 . . . 4  |-  ( b  =  B  ->  ( X  =  U. b  <->  X  =  Y ) )
27 raleq 2749 . . . 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 26367 . . 3  |-  Ref  =  { <. a ,  b
>.  |  ( U. a  =  U. b  /\  A. x  e.  b  E. y  e.  a  x  C_  y ) }
3023, 28, 29brabg 4300 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   U.cuni 3843   class class class wbr 4039   Refcref 26363
This theorem is referenced by:  refbas  26383  refssex  26384  ssref  26386  refref  26388  reftr  26392  fnessref  26396  refssfne  26397
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-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-xp 4711  df-rel 4712  df-ref 26367
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