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Theorem ssref 26363
Description: A subcover is a refinement of the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.)
Hypotheses
Ref Expression
ssref.1  |-  X  = 
U. A
ssref.2  |-  Y  = 
U. B
Assertion
Ref Expression
ssref  |-  ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  ->  B Ref A )

Proof of Theorem ssref
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqcom 2438 . . . 4  |-  ( X  =  Y  <->  Y  =  X )
21biimpi 187 . . 3  |-  ( X  =  Y  ->  Y  =  X )
323ad2ant3 980 . 2  |-  ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  ->  Y  =  X )
4 ssel2 3343 . . . . 5  |-  ( ( A  C_  B  /\  x  e.  A )  ->  x  e.  B )
543ad2antl2 1120 . . . 4  |-  ( ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  /\  x  e.  A
)  ->  x  e.  B )
6 ssid 3367 . . . 4  |-  x  C_  x
7 sseq2 3370 . . . . 5  |-  ( y  =  x  ->  (
x  C_  y  <->  x  C_  x
) )
87rspcev 3052 . . . 4  |-  ( ( x  e.  B  /\  x  C_  x )  ->  E. y  e.  B  x  C_  y )
95, 6, 8sylancl 644 . . 3  |-  ( ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  /\  x  e.  A
)  ->  E. y  e.  B  x  C_  y
)
109ralrimiva 2789 . 2  |-  ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  ->  A. x  e.  A  E. y  e.  B  x  C_  y
)
11 ssref.2 . . . 4  |-  Y  = 
U. B
12 ssref.1 . . . 4  |-  X  = 
U. A
1311, 12isref 26359 . . 3  |-  ( A  e.  C  ->  ( B Ref A  <->  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y
) ) )
14133ad2ant1 978 . 2  |-  ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  ->  ( B Ref A  <->  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y
) ) )
153, 10, 14mpbir2and 889 1  |-  ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  ->  B Ref A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    C_ wss 3320   U.cuni 4015   class class class wbr 4212   Refcref 26340
This theorem is referenced by:  refssfne  26374
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-ref 26344
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