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

Theorem ssref 26253
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 2406 . . . 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 3303 . . . . 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 3327 . . . 4  |-  x  C_  x
7 sseq2 3330 . . . . 5  |-  ( y  =  x  ->  (
x  C_  y  <->  x  C_  x
) )
87rspcev 3012 . . . 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 2749 . 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 26249 . . 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 1649    e. wcel 1721   A.wral 2666   E.wrex 2667    C_ wss 3280   U.cuni 3975   class class class wbr 4172   Refcref 26230
This theorem is referenced by:  refssfne  26264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-ref 26234
  Copyright terms: Public domain W3C validator