Theorem elrnmpt1 4896

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 2755	. . . 4 |- x e. _V
2		id 19	. . . . . . 7 |- ( x = z -> x = z )
3		csbeq1a 3081	. . . . . . 7 |- ( x = z -> A = [_ z / x ]_ A )
4	2, 3	eleq12d 2260	. . . . . 6 |- ( x = z -> ( x e. A <-> z e. [_ z / x ]_ A ) )
5		csbeq1a 3081	. . . . . . 7 |- ( x = z -> B = [_ z / x ]_ B )
6	5	biantrud 304	. . . . . 6 |- ( x = z -> ( z e. [_ z / x ]_ A <-> ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) ) )
7	4, 6	bitr2d 189	. . . . 5 |- ( x = z -> ( ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) <-> x e. A ) )
8	7	equcoms 1719	. . . 4 |- ( z = x -> ( ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) <-> x e. A ) )
9	1, 8	spcev 2847	. . 3 |- ( x e. A -> E. z ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) )
10		df-rex 2474	. . . . . 6 |- ( E. x e. A y = B <-> E. x ( x e. A /\ y = B ) )
11		nfv 1539	. . . . . . 7 |- F/ z ( x e. A /\ y = B )
12		nfcsb1v 3105	. . . . . . . . 9 |- F/_ x [_ z / x ]_ A
13	12	nfcri 2326	. . . . . . . 8 |- F/ x z e. [_ z / x ]_ A
14		nfcsb1v 3105	. . . . . . . . 9 |- F/_ x [_ z / x ]_ B
15	14	nfeq2 2344	. . . . . . . 8 |- F/ x y = [_ z / x ]_ B
16	13, 15	nfan 1576	. . . . . . 7 |- F/ x ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B )
17	5	eqeq2d 2201	. . . . . . . 8 |- ( x = z -> ( y = B <-> y = [_ z / x ]_ B ) )
18	4, 17	anbi12d 473	. . . . . . 7 |- ( x = z -> ( ( x e. A /\ y = B ) <-> ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B ) ) )
19	11, 16, 18	cbvex 1767	. . . . . 6 |- ( E. x ( x e. A /\ y = B ) <-> E. z ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B ) )
20	10, 19	bitri 184	. . . . 5 |- ( E. x e. A y = B <-> E. z ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B ) )
21		eqeq1 2196	. . . . . . 7 |- ( y = B -> ( y = [_ z / x ]_ B <-> B = [_ z / x ]_ B ) )
22	21	anbi2d 464	. . . . . 6 |- ( y = B -> ( ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B ) <-> ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) ) )
23	22	exbidv 1836	. . . . 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	bitrid 192	. . . 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 4893	. . . 4 |- ran F = { y | E. x e. A y = B }
27	24, 26	elab2g 2899	. . 3 |- ( B e. V -> ( B e. ran F <-> E. z ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) ) )
28	9, 27	imbitrrid 156	. 2 |- ( B e. V -> ( x e. A -> B e. ran F ) )
29	28	impcom 125	1 |- ( ( x e. A /\ B e. V ) -> B e. ran F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2160   E.wrex 2469   [_csb 3072   |-> cmpt 4079   ran crn 4645

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-mpt 4081  df-cnv 4652  df-dm 4654  df-rn 4655

This theorem is referenced by:  fliftel1  5816