Theorem abrexss 6352

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 2581	. . . 4 |- F/ x A. x e. A B e. C
2		abrexss.1	. . . . 5 |- F/_ x C
3	2	nfcri 2386	. . . 4 |- F/ x z e. C
4		eleq1 2301	. . . 4 |- ( z = B -> ( z e. C <-> B e. C ) )
5		vex 2824	. . . . 5 |- z e. _V
6	5	a1i 9	. . . 4 |- ( A. x e. A B e. C -> z e. _V )
7		rspa 2598	. . . 4 |- ( ( A. x e. A B e. C /\ x e. A ) -> B e. C )
8	1, 3, 4, 6, 7	elabreximd 6350	. . 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 3254	1 |- ( A. x e. A B e. C -> { y | E. x e. A y = B } C_ C )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   {cab 2224   F/_wnfc 2379   A.wral 2528   E.wrex 2529   _Vcvv 2821   C_ wss 3220

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233

This theorem is referenced by:  funimass4f  6353