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Theorem ssref 20527
Description: A subcover is a refinement of the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
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 )  ->  A Ref B )

Proof of Theorem ssref
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqcom 2458 . . . 4  |-  ( X  =  Y  <->  Y  =  X )
21biimpi 198 . . 3  |-  ( X  =  Y  ->  Y  =  X )
323ad2ant3 1031 . 2  |-  ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  ->  Y  =  X )
4 ssel2 3427 . . . . 5  |-  ( ( A  C_  B  /\  x  e.  A )  ->  x  e.  B )
543ad2antl2 1171 . . . 4  |-  ( ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  /\  x  e.  A
)  ->  x  e.  B )
6 ssid 3451 . . . 4  |-  x  C_  x
7 sseq2 3454 . . . . 5  |-  ( y  =  x  ->  (
x  C_  y  <->  x  C_  x
) )
87rspcev 3150 . . . 4  |-  ( ( x  e.  B  /\  x  C_  x )  ->  E. y  e.  B  x  C_  y )
95, 6, 8sylancl 668 . . 3  |-  ( ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  /\  x  e.  A
)  ->  E. y  e.  B  x  C_  y
)
109ralrimiva 2802 . 2  |-  ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  ->  A. x  e.  A  E. y  e.  B  x  C_  y
)
11 ssref.1 . . . 4  |-  X  = 
U. A
12 ssref.2 . . . 4  |-  Y  = 
U. B
1311, 12isref 20524 . . 3  |-  ( A  e.  C  ->  ( A Ref B  <->  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y
) ) )
14133ad2ant1 1029 . 2  |-  ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  ->  ( A Ref B  <->  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y
) ) )
153, 10, 14mpbir2and 933 1  |-  ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y )  ->  A Ref B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   A.wral 2737   E.wrex 2738    C_ wss 3404   U.cuni 4198   class class class wbr 4402   Refcref 20517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-xp 4840  df-rel 4841  df-ref 20520
This theorem is referenced by:  cmpcref  28677  refssfne  31014
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