Theorem elabrex 5652

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 1335	. . . 4 |- T.
2		csbeq1a 3007	. . . . . . 7 |- ( x = z -> B = [_ z / x ]_ B )
3	2	equcoms 1684	. . . . . 6 |- ( z = x -> B = [_ z / x ]_ B )
4		a1tru 1347	. . . . . 6 |- ( z = x -> T. )
5	3, 4	2thd 174	. . . . 5 |- ( z = x -> ( B = [_ z / x ]_ B <-> T. ) )
6	5	rspcev 2784	. . . 4 |- ( ( x e. A /\ T. ) -> E. z e. A B = [_ z / x ]_ B )
7	1, 6	mpan2 421	. . 3 |- ( x e. A -> E. z e. A B = [_ z / x ]_ B )
8		elabrex.1	. . . 4 |- B e. _V
9		eqeq1 2144	. . . . 5 |- ( y = B -> ( y = [_ z / x ]_ B <-> B = [_ z / x ]_ B ) )
10	9	rexbidv 2436	. . . 4 |- ( y = B -> ( E. z e. A y = [_ z / x ]_ B <-> E. z e. A B = [_ z / x ]_ B ) )
11	8, 10	elab 2823	. . 3 |- ( B e. { y | E. z e. A y = [_ z / x ]_ B } <-> E. z e. A B = [_ z / x ]_ B )
12	7, 11	sylibr 133	. 2 |- ( x e. A -> B e. { y | E. z e. A y = [_ z / x ]_ B } )
13		nfv 1508	. . . 4 |- F/ z y = B
14		nfcsb1v 3030	. . . . 5 |- F/_ x [_ z / x ]_ B
15	14	nfeq2 2291	. . . 4 |- F/ x y = [_ z / x ]_ B
16	2	eqeq2d 2149	. . . 4 |- ( x = z -> ( y = B <-> y = [_ z / x ]_ B ) )
17	13, 15, 16	cbvrex 2649	. . 3 |- ( E. x e. A y = B <-> E. z e. A y = [_ z / x ]_ B )
18	17	abbii 2253	. 2 |- { y | E. x e. A y = B } = { y | E. z e. A y = [_ z / x ]_ B }
19	12, 18	eleqtrrdi 2231	1 |- ( x e. A -> B e. { y | E. x e. A y = B } )

Syntax hints:   -> wi 4   = wceq 1331   T. wtru 1332   e. wcel 1480   {cab 2123   E.wrex 2415   _Vcvv 2681   [_csb 2998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999

This theorem is referenced by:  eusvobj2  5753