Theorem elrnmpt1 4948

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 2779	. . . 4 |- x e. _V
2		id 19	. . . . . . 7 |- ( x = z -> x = z )
3		csbeq1a 3110	. . . . . . 7 |- ( x = z -> A = [_ z / x ]_ A )
4	2, 3	eleq12d 2278	. . . . . 6 |- ( x = z -> ( x e. A <-> z e. [_ z / x ]_ A ) )
5		csbeq1a 3110	. . . . . . 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 1732	. . . 4 |- ( z = x -> ( ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) <-> x e. A ) )
9	1, 8	spcev 2875	. . 3 |- ( x e. A -> E. z ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) )
10		df-rex 2492	. . . . . 6 |- ( E. x e. A y = B <-> E. x ( x e. A /\ y = B ) )
11		nfv 1552	. . . . . . 7 |- F/ z ( x e. A /\ y = B )
12		nfcsb1v 3134	. . . . . . . . 9 |- F/_ x [_ z / x ]_ A
13	12	nfcri 2344	. . . . . . . 8 |- F/ x z e. [_ z / x ]_ A
14		nfcsb1v 3134	. . . . . . . . 9 |- F/_ x [_ z / x ]_ B
15	14	nfeq2 2362	. . . . . . . 8 |- F/ x y = [_ z / x ]_ B
16	13, 15	nfan 1589	. . . . . . 7 |- F/ x ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B )
17	5	eqeq2d 2219	. . . . . . . 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 1780	. . . . . 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 2214	. . . . . . 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 1849	. . . . 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 4945	. . . 4 |- ran F = { y | E. x e. A y = B }
27	24, 26	elab2g 2927	. . 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 1373   E.wex 1516   e. wcel 2178   E.wrex 2487   [_csb 3101   |-> cmpt 4121   ran crn 4694

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-mpt 4123  df-cnv 4701  df-dm 4703  df-rn 4704

This theorem is referenced by:  fliftel1  5886