Theorem elrnmpti 4750

Description: Membership in the range of a function. (Contributed by NM, 30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
rnmpt.1	|- F = ( x e. A |-> B )
elrnmpti.2	|- B e. _V

Assertion

Ref	Expression
elrnmpti	|- ( C e. ran F <-> E. x e. A C = B )

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)   F( x)

Proof of Theorem elrnmpti

Step	Hyp	Ref	Expression
1		elrnmpti.2	. . 3 |- B e. _V
2	1	rgenw 2459	. 2 |- A. x e. A B e. _V
3		rnmpt.1	. . 3 |- F = ( x e. A |-> B )
4	3	elrnmptg 4749	. 2 |- ( A. x e. A B e. _V -> ( C e. ran F <-> E. x e. A C = B ) )
5	2, 4	ax-mp 7	1 |- ( C e. ran F <-> E. x e. A C = B )

Syntax hints:   <-> wb 104   = wceq 1312   e. wcel 1461   A.wral 2388   E.wrex 2389   _Vcvv 2655   |-> cmpt 3947   ran crn 4498

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-mpt 3949  df-cnv 4505  df-dm 4507  df-rn 4508

This theorem is referenced by:  elrest  11964