Theorem abrexss 6331

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.

Step	Hyp	Ref	Expression
1		nfra1 2575	. . . 4 |- F/ x A. x e. A B e. C
2		abrexss.1	. . . . 5 |- F/_ x C
3	2	nfcri 2380	. . . 4 |- F/ x z e. C
4		eleq1 2297	. . . 4 |- ( z = B -> ( z e. C <-> B e. C ) )
5		vex 2818	. . . . 5 |- z e. _V
6	5	a1i 9	. . . 4 |- ( A. x e. A B e. C -> z e. _V )
7		rspa 2592	. . . 4 |- ( ( A. x e. A B e. C /\ x e. A ) -> B e. C )
8	1, 3, 4, 6, 7	elabreximd 6329	. . 3 |- ( ( A. x e. A B e. C /\ z e. { y | E. x e. A y = B } ) -> z e. C )
9	8	ex 115	. 2 |- ( A. x e. A B e. C -> ( z e. { y | E. x e. A y = B } -> z e. C ) )
10	9	ssrdv 3248	1 |- ( A. x e. A B e. C -> { y | E. x e. A y = B } C_ C )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   {cab 2220   F/_wnfc 2373   A.wral 2522   E.wrex 2523   _Vcvv 2815   C_ wss 3214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-in 3220  df-ss 3227

This theorem is referenced by:  funimass4f  6332