Theorem elabrex 5957

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 1406	. . . 4 |- T.
2		csbeq1a 3156	. . . . . . 7 |- ( x = z -> B = [_ z / x ]_ B )
3	2	equcoms 1760	. . . . . 6 |- ( z = x -> B = [_ z / x ]_ B )
4		trud 1418	. . . . . 6 |- ( z = x -> T. )
5	3, 4	2thd 175	. . . . 5 |- ( z = x -> ( B = [_ z / x ]_ B <-> T. ) )
6	5	rspcev 2929	. . . 4 |- ( ( x e. A /\ T. ) -> E. z e. A B = [_ z / x ]_ B )
7	1, 6	mpan2 429	. . 3 |- ( x e. A -> E. z e. A B = [_ z / x ]_ B )
8		elabrex.1	. . . 4 |- B e. _V
9		eqeq1 2245	. . . . 5 |- ( y = B -> ( y = [_ z / x ]_ B <-> B = [_ z / x ]_ B ) )
10	9	rexbidv 2551	. . . 4 |- ( y = B -> ( E. z e. A y = [_ z / x ]_ B <-> E. z e. A B = [_ z / x ]_ B ) )
11	8, 10	elab 2970	. . 3 |- ( B e. { y | E. z e. A y = [_ z / x ]_ B } <-> E. z e. A B = [_ z / x ]_ B )
12	7, 11	sylibr 134	. 2 |- ( x e. A -> B e. { y | E. z e. A y = [_ z / x ]_ B } )
13		nfv 1581	. . . 4 |- F/ z y = B
14		nfcsb1v 3180	. . . . 5 |- F/_ x [_ z / x ]_ B
15	14	nfeq2 2404	. . . 4 |- F/ x y = [_ z / x ]_ B
16	2	eqeq2d 2250	. . . 4 |- ( x = z -> ( y = B <-> y = [_ z / x ]_ B ) )
17	13, 15, 16	cbvrex 2783	. . 3 |- ( E. x e. A y = B <-> E. z e. A y = [_ z / x ]_ B )
18	17	abbii 2354	. 2 |- { y | E. x e. A y = B } = { y | E. z e. A y = [_ z / x ]_ B }
19	12, 18	eleqtrrdi 2332	1 |- ( x e. A -> B e. { y | E. x e. A y = B } )

Syntax hints:   -> wi 4   = wceq 1402   T. wtru 1403   e. wcel 2209   {cab 2224   E.wrex 2529   _Vcvv 2821   [_csb 3147

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-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148

This theorem is referenced by:  eusvobj2  6065