Theorem elabrex 6143

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 1383	. . . 4 |- T.
2		csbeq1a 3444	. . . . . . 7 |- ( x = z -> B = [_ z / x ]_ B )
3	2	equcoms 1744	. . . . . 6 |- ( z = x -> B = [_ z / x ]_ B )
4		a1tru 1395	. . . . . 6 |- ( z = x -> T. )
5	3, 4	2thd 240	. . . . 5 |- ( z = x -> ( B = [_ z / x ]_ B <-> T. ) )
6	5	rspcev 3214	. . . 4 |- ( ( x e. A /\ T. ) -> E. z e. A B = [_ z / x ]_ B )
7	1, 6	mpan2 671	. . 3 |- ( x e. A -> E. z e. A B = [_ z / x ]_ B )
8		elabrex.1	. . . 4 |- B e. _V
9		eqeq1 2471	. . . . 5 |- ( y = B -> ( y = [_ z / x ]_ B <-> B = [_ z / x ]_ B ) )
10	9	rexbidv 2973	. . . 4 |- ( y = B -> ( E. z e. A y = [_ z / x ]_ B <-> E. z e. A B = [_ z / x ]_ B ) )
11	8, 10	elab 3250	. . 3 |- ( B e. { y | E. z e. A y = [_ z / x ]_ B } <-> E. z e. A B = [_ z / x ]_ B )
12	7, 11	sylibr 212	. 2 |- ( x e. A -> B e. { y | E. z e. A y = [_ z / x ]_ B } )
13		nfv 1683	. . . 4 |- F/ z y = B
14		nfcsb1v 3451	. . . . 5 |- F/_ x [_ z / x ]_ B
15	14	nfeq2 2646	. . . 4 |- F/ x y = [_ z / x ]_ B
16	2	eqeq2d 2481	. . . 4 |- ( x = z -> ( y = B <-> y = [_ z / x ]_ B ) )
17	13, 15, 16	cbvrex 3085	. . 3 |- ( E. x e. A y = B <-> E. z e. A y = [_ z / x ]_ B )
18	17	abbii 2601	. 2 |- { y | E. x e. A y = B } = { y | E. z e. A y = [_ z / x ]_ B }
19	12, 18	syl6eleqr 2566	1 |- ( x e. A -> B e. { y | E. x e. A y = B } )

Syntax hints:   -> wi 4   = wceq 1379   T. wtru 1380   e. wcel 1767   {cab 2452   E.wrex 2815   _Vcvv 3113   [_csb 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-v 3115  df-sbc 3332  df-csb 3436

This theorem is referenced by:  eusvobj2  6277  lss1d  17409  prdsxmetlem  20634  prdsbl  20757  itg2monolem1  21920  heibor1  29937  dihglblem5  36113