Theorem elrnmpt 4783

Description: The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.)

Hypothesis

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

Assertion

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

Distinct variable group:   x, C

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

Proof of Theorem elrnmpt

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2144	. . 3 |- ( y = C -> ( y = B <-> C = B ) )
2	1	rexbidv 2436	. 2 |- ( y = C -> ( E. x e. A y = B <-> E. x e. A C = B ) )
3		rnmpt.1	. . 3 |- F = ( x e. A |-> B )
4	3	rnmpt 4782	. 2 |- ran F = { y | E. x e. A y = B }
5	2, 4	elab2g 2826	1 |- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   E.wrex 2415   |-> 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-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:  elrnmpt1s  4784  elrnmptdv  4788  elrnmpt2d  4789  fifo  6861