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

Theorem refssex 26046
Description: Every set in a refinement has a superset in the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.)
Assertion
Ref Expression
refssex  |-  ( ( A Ref B  /\  S  e.  B )  ->  E. x  e.  A  S  C_  x )
Distinct variable groups:    x, A    x, S
Allowed substitution hint:    B( x)

Proof of Theorem refssex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 refrel 26043 . . . . 5  |-  Rel  Ref
21brrelex2i 4853 . . . 4  |-  ( A Ref B  ->  B  e.  _V )
3 eqid 2381 . . . . . 6  |-  U. A  =  U. A
4 eqid 2381 . . . . . 6  |-  U. B  =  U. B
53, 4isref 26044 . . . . 5  |-  ( B  e.  _V  ->  ( A Ref B  <->  ( U. A  =  U. B  /\  A. y  e.  B  E. x  e.  A  y  C_  x ) ) )
65simplbda 608 . . . 4  |-  ( ( B  e.  _V  /\  A Ref B )  ->  A. y  e.  B  E. x  e.  A  y  C_  x )
72, 6mpancom 651 . . 3  |-  ( A Ref B  ->  A. y  e.  B  E. x  e.  A  y  C_  x )
8 sseq1 3306 . . . . 5  |-  ( y  =  S  ->  (
y  C_  x  <->  S  C_  x
) )
98rexbidv 2664 . . . 4  |-  ( y  =  S  ->  ( E. x  e.  A  y  C_  x  <->  E. x  e.  A  S  C_  x
) )
109rspccv 2986 . . 3  |-  ( A. y  e.  B  E. x  e.  A  y  C_  x  ->  ( S  e.  B  ->  E. x  e.  A  S  C_  x
) )
117, 10syl 16 . 2  |-  ( A Ref B  ->  ( S  e.  B  ->  E. x  e.  A  S  C_  x ) )
1211imp 419 1  |-  ( ( A Ref B  /\  S  e.  B )  ->  E. x  e.  A  S  C_  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2643   E.wrex 2644   _Vcvv 2893    C_ wss 3257   U.cuni 3951   class class class wbr 4147   Refcref 26025
This theorem is referenced by:  reftr  26054  refssfne  26059
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-rab 2652  df-v 2895  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-br 4148  df-opab 4202  df-xp 4818  df-rel 4819  df-ref 26029
  Copyright terms: Public domain W3C validator