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Theorem isref 26340
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 26329. 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 26339 . . . . 5  |-  Rel  Ref
21brrelexi 4910 . . . 4  |-  ( A Ref B  ->  A  e.  _V )
32adantl 453 . . 3  |-  ( ( B  e.  C  /\  A Ref B )  ->  A  e.  _V )
4 simpl 444 . . 3  |-  ( ( B  e.  C  /\  A Ref B )  ->  B  e.  C )
53, 4jca 519 . 2  |-  ( ( B  e.  C  /\  A Ref B )  -> 
( A  e.  _V  /\  B  e.  C ) )
6 simpr 448 . . . . . . 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 2490 . . . . . 6  |-  ( ( B  e.  C  /\  X  =  Y )  ->  U. A  =  U. B )
10 uniexg 4698 . . . . . . 7  |-  ( B  e.  C  ->  U. B  e.  _V )
1110adantr 452 . . . . . 6  |-  ( ( B  e.  C  /\  X  =  Y )  ->  U. B  e.  _V )
129, 11eqeltrd 2509 . . . . 5  |-  ( ( B  e.  C  /\  X  =  Y )  ->  U. A  e.  _V )
13 uniexb 4744 . . . . 5  |-  ( A  e.  _V  <->  U. A  e. 
_V )
1412, 13sylibr 204 . . . 4  |-  ( ( B  e.  C  /\  X  =  Y )  ->  A  e.  _V )
1514adantrr 698 . . 3  |-  ( ( B  e.  C  /\  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y ) )  ->  A  e.  _V )
16 simpl 444 . . 3  |-  ( ( B  e.  C  /\  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y ) )  ->  B  e.  C
)
1715, 16jca 519 . 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 4016 . . . . . 6  |-  ( a  =  A  ->  U. a  =  U. A )
1918, 7syl6eqr 2485 . . . . 5  |-  ( a  =  A  ->  U. a  =  X )
2019eqeq1d 2443 . . . 4  |-  ( a  =  A  ->  ( U. a  =  U. b 
<->  X  =  U. b
) )
21 rexeq 2897 . . . . 5  |-  ( a  =  A  ->  ( E. y  e.  a  x  C_  y  <->  E. y  e.  A  x  C_  y
) )
2221ralbidv 2717 . . . 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 692 . . 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 4016 . . . . . 6  |-  ( b  =  B  ->  U. b  =  U. B )
2524, 8syl6eqr 2485 . . . . 5  |-  ( b  =  B  ->  U. b  =  Y )
2625eqeq2d 2446 . . . 4  |-  ( b  =  B  ->  ( X  =  U. b  <->  X  =  Y ) )
27 raleq 2896 . . . 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 692 . . 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 26325 . . 3  |-  Ref  =  { <. a ,  b
>.  |  ( U. a  =  U. b  /\  A. x  e.  b  E. y  e.  a  x  C_  y ) }
3023, 28, 29brabg 4466 . 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 869 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   _Vcvv 2948    C_ wss 3312   U.cuni 4007   class class class wbr 4204   Refcref 26321
This theorem is referenced by:  refbas  26341  refssex  26342  ssref  26344  refref  26346  reftr  26350  fnessref  26354  refssfne  26355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-ref 26325
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