Theorem elrnmpt1 4862

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 2733	. . . 4 |- x e. _V
2		id 19	. . . . . . 7 |- ( x = z -> x = z )
3		csbeq1a 3058	. . . . . . 7 |- ( x = z -> A = [_ z / x ]_ A )
4	2, 3	eleq12d 2241	. . . . . 6 |- ( x = z -> ( x e. A <-> z e. [_ z / x ]_ A ) )
5		csbeq1a 3058	. . . . . . 7 |- ( x = z -> B = [_ z / x ]_ B )
6	5	biantrud 302	. . . . . 6 |- ( x = z -> ( z e. [_ z / x ]_ A <-> ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) ) )
7	4, 6	bitr2d 188	. . . . 5 |- ( x = z -> ( ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) <-> x e. A ) )
8	7	equcoms 1701	. . . 4 |- ( z = x -> ( ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) <-> x e. A ) )
9	1, 8	spcev 2825	. . 3 |- ( x e. A -> E. z ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) )
10		df-rex 2454	. . . . . 6 |- ( E. x e. A y = B <-> E. x ( x e. A /\ y = B ) )
11		nfv 1521	. . . . . . 7 |- F/ z ( x e. A /\ y = B )
12		nfcsb1v 3082	. . . . . . . . 9 |- F/_ x [_ z / x ]_ A
13	12	nfcri 2306	. . . . . . . 8 |- F/ x z e. [_ z / x ]_ A
14		nfcsb1v 3082	. . . . . . . . 9 |- F/_ x [_ z / x ]_ B
15	14	nfeq2 2324	. . . . . . . 8 |- F/ x y = [_ z / x ]_ B
16	13, 15	nfan 1558	. . . . . . 7 |- F/ x ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B )
17	5	eqeq2d 2182	. . . . . . . 8 |- ( x = z -> ( y = B <-> y = [_ z / x ]_ B ) )
18	4, 17	anbi12d 470	. . . . . . 7 |- ( x = z -> ( ( x e. A /\ y = B ) <-> ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B ) ) )
19	11, 16, 18	cbvex 1749	. . . . . 6 |- ( E. x ( x e. A /\ y = B ) <-> E. z ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B ) )
20	10, 19	bitri 183	. . . . 5 |- ( E. x e. A y = B <-> E. z ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B ) )
21		eqeq1 2177	. . . . . . 7 |- ( y = B -> ( y = [_ z / x ]_ B <-> B = [_ z / x ]_ B ) )
22	21	anbi2d 461	. . . . . 6 |- ( y = B -> ( ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B ) <-> ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) ) )
23	22	exbidv 1818	. . . . 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 191	. . . 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 4859	. . . 4 |- ran F = { y | E. x e. A y = B }
27	24, 26	elab2g 2877	. . 3 |- ( B e. V -> ( B e. ran F <-> E. z ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) ) )
28	9, 27	syl5ibr 155	. 2 |- ( B e. V -> ( x e. A -> B e. ran F ) )
29	28	impcom 124	1 |- ( ( x e. A /\ B e. V ) -> B e. ran F )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141   E.wrex 2449   [_csb 3049   |-> cmpt 4050   ran crn 4612

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-mpt 4052  df-cnv 4619  df-dm 4621  df-rn 4622

This theorem is referenced by:  fliftel1  5773