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Theorem abrexss 23040
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 2593 . . . 4  |-  F/ x A. x  e.  A  B  e.  C
2 nfcv 2419 . . . . 5  |-  F/_ x
z
3 abrexss.1 . . . . 5  |-  F/_ x C
42, 3nfel 2427 . . . 4  |-  F/ x  z  e.  C
5 eleq1 2343 . . . 4  |-  ( z  =  B  ->  (
z  e.  C  <->  B  e.  C ) )
6 vex 2791 . . . . 5  |-  z  e. 
_V
76a1i 10 . . . 4  |-  ( A. x  e.  A  B  e.  C  ->  z  e. 
_V )
8 id 19 . . . . 5  |-  ( A. x  e.  A  B  e.  C  ->  A. x  e.  A  B  e.  C )
98r19.21bi 2641 . . . 4  |-  ( ( A. x  e.  A  B  e.  C  /\  x  e.  A )  ->  B  e.  C )
101, 4, 5, 7, 9elabreximd 23039 . . 3  |-  ( ( A. x  e.  A  B  e.  C  /\  z  e.  { y  |  E. x  e.  A  y  =  B }
)  ->  z  e.  C )
1110ex 423 . 2  |-  ( A. x  e.  A  B  e.  C  ->  ( z  e.  { y  |  E. x  e.  A  y  =  B }  ->  z  e.  C ) )
1211ssrdv 3185 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 1623    e. wcel 1684   {cab 2269   F/_wnfc 2406   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152
This theorem is referenced by:  funimass4f  23042  measvunilem  23540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166
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