Theorem elrnmpt1 4880

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 2742	. . . 4 |- x e. _V
2		id 19	. . . . . . 7 |- ( x = z -> x = z )
3		csbeq1a 3068	. . . . . . 7 |- ( x = z -> A = [_ z / x ]_ A )
4	2, 3	eleq12d 2248	. . . . . 6 |- ( x = z -> ( x e. A <-> z e. [_ z / x ]_ A ) )
5		csbeq1a 3068	. . . . . . 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 1708	. . . 4 |- ( z = x -> ( ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) <-> x e. A ) )
9	1, 8	spcev 2834	. . 3 |- ( x e. A -> E. z ( z e. [_ z / x ]_ A /\ B = [_ z / x ]_ B ) )
10		df-rex 2461	. . . . . 6 |- ( E. x e. A y = B <-> E. x ( x e. A /\ y = B ) )
11		nfv 1528	. . . . . . 7 |- F/ z ( x e. A /\ y = B )
12		nfcsb1v 3092	. . . . . . . . 9 |- F/_ x [_ z / x ]_ A
13	12	nfcri 2313	. . . . . . . 8 |- F/ x z e. [_ z / x ]_ A
14		nfcsb1v 3092	. . . . . . . . 9 |- F/_ x [_ z / x ]_ B
15	14	nfeq2 2331	. . . . . . . 8 |- F/ x y = [_ z / x ]_ B
16	13, 15	nfan 1565	. . . . . . 7 |- F/ x ( z e. [_ z / x ]_ A /\ y = [_ z / x ]_ B )
17	5	eqeq2d 2189	. . . . . . . 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 1756	. . . . . 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 2184	. . . . . . 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 1825	. . . . 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 4877	. . . 4 |- ran F = { y | E. x e. A y = B }
27	24, 26	elab2g 2886	. . 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 1353   E.wex 1492   e. wcel 2148   E.wrex 2456   [_csb 3059   |-> cmpt 4066   ran crn 4629

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-mpt 4068  df-cnv 4636  df-dm 4638  df-rn 4639

This theorem is referenced by:  fliftel1  5797