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Theorem ssref 26386
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 2298 . . . 4  |-  ( X  =  Y  <->  Y  =  X )
21biimpi 186 . . 3  |-  ( X  =  Y  ->  Y  =  X )
323ad2ant3 978 . 2  |-  ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  ->  Y  =  X )
4 ssel2 3188 . . . . 5  |-  ( ( A  C_  B  /\  x  e.  A )  ->  x  e.  B )
543ad2antl2 1118 . . . 4  |-  ( ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  /\  x  e.  A
)  ->  x  e.  B )
6 ssid 3210 . . . 4  |-  x  C_  x
7 sseq2 3213 . . . . 5  |-  ( y  =  x  ->  (
x  C_  y  <->  x  C_  x
) )
87rspcev 2897 . . . 4  |-  ( ( x  e.  B  /\  x  C_  x )  ->  E. y  e.  B  x  C_  y )
95, 6, 8sylancl 643 . . 3  |-  ( ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  /\  x  e.  A
)  ->  E. y  e.  B  x  C_  y
)
109ralrimiva 2639 . 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 26382 . . 3  |-  ( A  e.  C  ->  ( B Ref A  <->  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y
) ) )
14133ad2ant1 976 . 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 888 1  |-  ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  ->  B Ref A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   U.cuni 3843   class class class wbr 4039   Refcref 26363
This theorem is referenced by:  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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