MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  refssex Structured version   Unicode version

Theorem refssex 19878
Description: Every set in a refinement has a superset in the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Assertion
Ref Expression
refssex  |-  ( ( A Ref B  /\  S  e.  A )  ->  E. x  e.  B  S  C_  x )
Distinct variable groups:    x, B    x, S
Allowed substitution hint:    A( x)

Proof of Theorem refssex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 refrel 19875 . . . . 5  |-  Rel  Ref
21brrelexi 5026 . . . 4  |-  ( A Ref B  ->  A  e.  _V )
3 eqid 2441 . . . . . 6  |-  U. A  =  U. A
4 eqid 2441 . . . . . 6  |-  U. B  =  U. B
53, 4isref 19876 . . . . 5  |-  ( A  e.  _V  ->  ( A Ref B  <->  ( U. B  =  U. A  /\  A. y  e.  A  E. x  e.  B  y  C_  x ) ) )
65simplbda 624 . . . 4  |-  ( ( A  e.  _V  /\  A Ref B )  ->  A. y  e.  A  E. x  e.  B  y  C_  x )
72, 6mpancom 669 . . 3  |-  ( A Ref B  ->  A. y  e.  A  E. x  e.  B  y  C_  x )
8 sseq1 3507 . . . . 5  |-  ( y  =  S  ->  (
y  C_  x  <->  S  C_  x
) )
98rexbidv 2952 . . . 4  |-  ( y  =  S  ->  ( E. x  e.  B  y  C_  x  <->  E. x  e.  B  S  C_  x
) )
109rspccv 3191 . . 3  |-  ( A. y  e.  A  E. x  e.  B  y  C_  x  ->  ( S  e.  A  ->  E. x  e.  B  S  C_  x
) )
117, 10syl 16 . 2  |-  ( A Ref B  ->  ( S  e.  A  ->  E. x  e.  B  S  C_  x ) )
1211imp 429 1  |-  ( ( A Ref B  /\  S  e.  A )  ->  E. x  e.  B  S  C_  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   A.wral 2791   E.wrex 2792   _Vcvv 3093    C_ wss 3458   U.cuni 4230   class class class wbr 4433   Refcref 19869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-xp 4991  df-rel 4992  df-ref 19872
This theorem is referenced by:  reftr  19881  refun0  19882  refssfne  30144
  Copyright terms: Public domain W3C validator