Theorem elrnmpti 4787

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 2485	. 2 |- A. x e. A B e. _V
3		rnmpt.1	. . 3 |- F = ( x e. A |-> B )
4	3	elrnmptg 4786	. 2 |- ( A. x e. A B e. _V -> ( C e. ran F <-> E. x e. A C = B ) )
5	2, 4	ax-mp 5	1 |- ( C e. ran F <-> E. x e. A C = B )

Syntax hints:   <-> wb 104   = wceq 1331   e. wcel 1480   A.wral 2414   E.wrex 2415   _Vcvv 2681   |-> cmpt 3984   ran crn 4535

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-mpt 3986  df-cnv 4542  df-dm 4544  df-rn 4545

This theorem is referenced by:  elrest  12116