Theorem elrnmpt1 4585

Description: Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
rnmpt.1	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
elrnmpt1	|- ( ( x e. A /\ B e. V ) -> B e. ran F )

Proof of Theorem elrnmpt1

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2560	. . . 4 |- x e. _V
2		id 19	. . . . . . 7 |- ( x = z -> x = z )
3		csbeq1a 2860	. . . . . . 7 |- ( x = z -> A = [_ z / x ]_ A )
4	2, 3	eleq12d 2108	. . . . . 6 |- ( x = z -> ( x e. A <-> z e. [_ z / x ]_ A ) )
5		csbeq1a 2860	. . . . . . 7 |- ( x = z -> B = [_ z / x ]_ B )
6	5	biantrud 288	. . . . . 6 |- ( x = z -> ( z e. [_ z / x ]_ A <-> ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) ) )
7	4, 6	bitr2d 178	. . . . 5 |- ( x = z -> ( ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) <-> x e. A ) )
8	7	equcoms 1594	. . . 4 |- ( z = x -> ( ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) <-> x e. A ) )
9	1, 8	spcev 2647	. . 3 |- ( x e. A -> E. z ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) )
10		df-rex 2312	. . . . . 6 |- ( E. x e. A y = B <-> E. x ( x e. A /\ y = B ) )
11		nfv 1421	. . . . . . 7 |- F/ z ( x e. A /\ y = B )
12		nfcsb1v 2882	. . . . . . . . 9 |- F/_ x [_ z / x ]_ A
13	12	nfcri 2172	. . . . . . . 8 |- F/ x z e. [_ z / x ]_ A
14		nfcsb1v 2882	. . . . . . . . 9 |- F/_ x [_ z / x ]_ B
15	14	nfeq2 2189	. . . . . . . 8 |- F/ x y = [_ z / x ]_ B
16	13, 15	nfan 1457	. . . . . . 7 |- F/ x ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B )
17	5	eqeq2d 2051	. . . . . . . 8 |- ( x = z -> ( y = B <-> y = [_ z / x ]_ B ) )
18	4, 17	anbi12d 442	. . . . . . 7 |- ( x = z -> ( ( x e. A /\ y = B ) <-> ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B ) ) )
19	11, 16, 18	cbvex 1639	. . . . . 6 |- ( E. x ( x e. A /\ y = B ) <-> E. z ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B ) )
20	10, 19	bitri 173	. . . . 5 |- ( E. x e. A y = B <-> E. z ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B ) )
21		eqeq1 2046	. . . . . . 7 |- ( y = B -> ( y = [_ z / x ]_ B <-> B = [_ z / x ]_ B ) )
22	21	anbi2d 437	. . . . . 6 |- ( y = B -> ( ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B ) <-> ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) ) )
23	22	exbidv 1706	. . . . 5 |- ( y = B -> ( E. z ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B ) <-> E. z ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) ) )
24	20, 23	syl5bb 181	. . . 4 |- ( y = B -> ( E. x e. A y = B <-> E. z ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) ) )
25		rnmpt.1	. . . . 5 |- F = ( x e. A |-> B )
26	25	rnmpt 4582	. . . 4 |- ran F = { y | E. x e. A y = B }
27	24, 26	elab2g 2689	. . 3 |- ( B e. V -> ( B e. ran F <-> E. z ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) ) )
28	9, 27	syl5ibr 145	. 2 |- ( B e. V -> ( x e. A -> B e. ran F ) )
29	28	impcom 116	1 |- ( ( x e. A /\ B e. V ) -> B e. ran F )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   E.wrex 2307   [_csb 2852   |-> cmpt 3818   ran crn 4346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-mpt 3820  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by:  fliftel1  5434