Theorem elabrex 5418

Description: Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)

Hypothesis

Ref	Expression
elabrex.1	|- B e. _V

Assertion

Ref	Expression
elabrex	|- ( x e. A -> B e. { y | E. x e. A y = B } )

Distinct variable groups:   y, B   x, y, A

Allowed substitution hint:   B( x)

Proof of Theorem elabrex

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tru 1288	. . . 4 |- T.
2		csbeq1a 2916	. . . . . . 7 |- ( x = z -> B = [_ z / x ]_ B )
3	2	equcoms 1634	. . . . . 6 |- ( z = x -> B = [_ z / x ]_ B )
4		a1tru 1300	. . . . . 6 |- ( z = x -> T. )
5	3, 4	2thd 173	. . . . 5 |- ( z = x -> ( B = [_ z / x ]_ B <-> T. ) )
6	5	rspcev 2701	. . . 4 |- ( ( x e. A /\ T. ) -> E. z e. A B = [_ z / x ]_ B )
7	1, 6	mpan2 415	. . 3 |- ( x e. A -> E. z e. A B = [_ z / x ]_ B )
8		elabrex.1	. . . 4 |- B e. _V
9		eqeq1 2087	. . . . 5 |- ( y = B -> ( y = [_ z / x ]_ B <-> B = [_ z / x ]_ B ) )
10	9	rexbidv 2369	. . . 4 |- ( y = B -> ( E. z e. A y = [_ z / x ]_ B <-> E. z e. A B = [_ z / x ]_ B ) )
11	8, 10	elab 2738	. . 3 |- ( B e. { y | E. z e. A y = [_ z / x ]_ B } <-> E. z e. A B = [_ z / x ]_ B )
12	7, 11	sylibr 132	. 2 |- ( x e. A -> B e. { y | E. z e. A y = [_ z / x ]_ B } )
13		nfv 1461	. . . 4 |- F/ z y = B
14		nfcsb1v 2938	. . . . 5 |- F/_ x [_ z / x ]_ B
15	14	nfeq2 2230	. . . 4 |- F/ x y = [_ z / x ]_ B
16	2	eqeq2d 2092	. . . 4 |- ( x = z -> ( y = B <-> y = [_ z / x ]_ B ) )
17	13, 15, 16	cbvrex 2574	. . 3 |- ( E. x e. A y = B <-> E. z e. A y = [_ z / x ]_ B )
18	17	abbii 2194	. 2 |- { y | E. x e. A y = B } = { y | E. z e. A y = [_ z / x ]_ B }
19	12, 18	syl6eleqr 2172	1 |- ( x e. A -> B e. { y | E. x e. A y = B } )

Syntax hints:   -> wi 4   = wceq 1284   T. wtru 1285   e. wcel 1433   {cab 2067   E.wrex 2349   _Vcvv 2601   [_csb 2908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909

This theorem is referenced by:  eusvobj2  5518