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

Theorem refssex 28724
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 28721 . . . . 5  |-  Rel  Ref
21brrelex2i 4991 . . . 4  |-  ( A Ref B  ->  B  e.  _V )
3 eqid 2454 . . . . . 6  |-  U. A  =  U. A
4 eqid 2454 . . . . . 6  |-  U. B  =  U. B
53, 4isref 28722 . . . . 5  |-  ( B  e.  _V  ->  ( A Ref B  <->  ( U. A  =  U. B  /\  A. y  e.  B  E. x  e.  A  y  C_  x ) ) )
65simplbda 624 . . . 4  |-  ( ( B  e.  _V  /\  A Ref B )  ->  A. y  e.  B  E. x  e.  A  y  C_  x )
72, 6mpancom 669 . . 3  |-  ( A Ref B  ->  A. y  e.  B  E. x  e.  A  y  C_  x )
8 sseq1 3488 . . . . 5  |-  ( y  =  S  ->  (
y  C_  x  <->  S  C_  x
) )
98rexbidv 2868 . . . 4  |-  ( y  =  S  ->  ( E. x  e.  A  y  C_  x  <->  E. x  e.  A  S  C_  x
) )
109rspccv 3176 . . 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 429 1  |-  ( ( A Ref B  /\  S  e.  B )  ->  E. x  e.  A  S  C_  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   E.wrex 2800   _Vcvv 3078    C_ wss 3439   U.cuni 4202   class class class wbr 4403   Refcref 28703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-ref 28707
This theorem is referenced by:  reftr  28732  refssfne  28737
  Copyright terms: Public domain W3C validator