Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ssref Unicode version

Theorem ssref 26283
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 2285 . . . 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 3175 . . . . 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 3197 . . . 4  |-  x  C_  x
7 sseq2 3200 . . . . 5  |-  ( y  =  x  ->  (
x  C_  y  <->  x  C_  x
) )
87rspcev 2884 . . . 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 2626 . 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 26279 . . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   U.cuni 3827   class class class wbr 4023   Refcref 26260
This theorem is referenced by:  refssfne  26294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-ref 26264
  Copyright terms: Public domain W3C validator