Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  abrexss Structured version   Unicode version

Theorem abrexss 23993
Description: A necessary condition for an image set to be a subset. (Contributed by Thierry Arnoux, 6-Feb-2017.)
Hypothesis
Ref Expression
abrexss.1  |-  F/_ x C
Assertion
Ref Expression
abrexss  |-  ( A. x  e.  A  B  e.  C  ->  { y  |  E. x  e.  A  y  =  B }  C_  C )
Distinct variable groups:    x, y    y, A    y, B
Allowed substitution hints:    A( x)    B( x)    C( x, y)

Proof of Theorem abrexss
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfra1 2756 . . . 4  |-  F/ x A. x  e.  A  B  e.  C
2 abrexss.1 . . . . 5  |-  F/_ x C
32nfcri 2566 . . . 4  |-  F/ x  z  e.  C
4 eleq1 2496 . . . 4  |-  ( z  =  B  ->  (
z  e.  C  <->  B  e.  C ) )
5 vex 2959 . . . . 5  |-  z  e. 
_V
65a1i 11 . . . 4  |-  ( A. x  e.  A  B  e.  C  ->  z  e. 
_V )
7 id 20 . . . . 5  |-  ( A. x  e.  A  B  e.  C  ->  A. x  e.  A  B  e.  C )
87r19.21bi 2804 . . . 4  |-  ( ( A. x  e.  A  B  e.  C  /\  x  e.  A )  ->  B  e.  C )
91, 3, 4, 6, 8elabreximd 23991 . . 3  |-  ( ( A. x  e.  A  B  e.  C  /\  z  e.  { y  |  E. x  e.  A  y  =  B }
)  ->  z  e.  C )
109ex 424 . 2  |-  ( A. x  e.  A  B  e.  C  ->  ( z  e.  { y  |  E. x  e.  A  y  =  B }  ->  z  e.  C ) )
1110ssrdv 3354 1  |-  ( A. x  e.  A  B  e.  C  ->  { y  |  E. x  e.  A  y  =  B }  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {cab 2422   F/_wnfc 2559   A.wral 2705   E.wrex 2706   _Vcvv 2956    C_ wss 3320
This theorem is referenced by:  funimass4f  24044  measvunilem  24566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-in 3327  df-ss 3334
  Copyright terms: Public domain W3C validator