Theorem elabrex 6100

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 1441	. . . 4 |- T.
2		csbeq1a 3340	. . . . . . 7 |- ( x = z -> B = [_ z / x ]_ B )
3	2	equcoms 1849	. . . . . 6 |- ( z = x -> B = [_ z / x ]_ B )
4		a1tru 1453	. . . . . 6 |- ( z = x -> T. )
5	3, 4	2thd 243	. . . . 5 |- ( z = x -> ( B = [_ z / x ]_ B <-> T. ) )
6	5	rspcev 3118	. . . 4 |- ( ( x e. A /\ T. ) -> E. z e. A B = [_ z / x ]_ B )
7	1, 6	mpan2 675	. . 3 |- ( x e. A -> E. z e. A B = [_ z / x ]_ B )
8		elabrex.1	. . . 4 |- B e. _V
9		eqeq1 2426	. . . . 5 |- ( y = B -> ( y = [_ z / x ]_ B <-> B = [_ z / x ]_ B ) )
10	9	rexbidv 2872	. . . 4 |- ( y = B -> ( E. z e. A y = [_ z / x ]_ B <-> E. z e. A B = [_ z / x ]_ B ) )
11	8, 10	elab 3153	. . 3 |- ( B e. { y | E. z e. A y = [_ z / x ]_ B } <-> E. z e. A B = [_ z / x ]_ B )
12	7, 11	sylibr 215	. 2 |- ( x e. A -> B e. { y | E. z e. A y = [_ z / x ]_ B } )
13		nfv 1755	. . . 4 |- F/ z y = B
14		nfcsb1v 3347	. . . . 5 |- F/_ x [_ z / x ]_ B
15	14	nfeq2 2578	. . . 4 |- F/ x y = [_ z / x ]_ B
16	2	eqeq2d 2432	. . . 4 |- ( x = z -> ( y = B <-> y = [_ z / x ]_ B ) )
17	13, 15, 16	cbvrex 2987	. . 3 |- ( E. x e. A y = B <-> E. z e. A y = [_ z / x ]_ B )
18	17	abbii 2538	. 2 |- { y | E. x e. A y = B } = { y | E. z e. A y = [_ z / x ]_ B }
19	12, 18	syl6eleqr 2511	1 |- ( x e. A -> B e. { y | E. x e. A y = B } )

Syntax hints:   -> wi 4   = wceq 1437   T. wtru 1438   e. wcel 1872   {cab 2408   E.wrex 2709   _Vcvv 3016   [_csb 3331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ral 2713  df-rex 2714  df-v 3018  df-sbc 3236  df-csb 3332

This theorem is referenced by:  eusvobj2  6235  lss1d  18122  prdsxmetlem  21318  prdsbl  21441  itg2monolem1  22643  heibor1  32043  dihglblem5  34772