Theorem abrexss 29350

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 2941	. . . 4 |- F/ x A. x e. A B e. C
2		abrexss.1	. . . . 5 |- F/_ x C
3	2	nfcri 2758	. . . 4 |- F/ x z e. C
4		eleq1 2689	. . . 4 |- ( z = B -> ( z e. C <-> B e. C ) )
5		vex 3203	. . . . 5 |- z e. _V
6	5	a1i 11	. . . 4 |- ( A. x e. A B e. C -> z e. _V )
7		rspa 2930	. . . 4 |- ( ( A. x e. A B e. C /\ x e. A ) -> B e. C )
8	1, 3, 4, 6, 7	elabreximd 29348	. . 3 |- ( ( A. x e. A B e. C /\ z e. { y | E. x e. A y = B } ) -> z e. C )
9	8	ex 450	. 2 |- ( A. x e. A B e. C -> ( z e. { y | E. x e. A y = B } -> z e. C ) )
10	9	ssrdv 3609	1 |- ( A. x e. A B e. C -> { y | E. x e. A y = B } C_ C )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588

This theorem is referenced by:  funimass4f  29437  measvunilem  30275