Theorem elrnmpt 4680

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 2094	. . 3 |- ( y = C -> ( y = B <-> C = B ) )
2	1	rexbidv 2381	. 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 4679	. 2 |- ran F = { y | E. x e. A y = B }
5	2, 4	elab2g 2762	1 |- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   e. wcel 1438   E.wrex 2360   |-> cmpt 3897   ran crn 4437

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-br 3844  df-opab 3898  df-mpt 3899  df-cnv 4444  df-dm 4446  df-rn 4447

This theorem is referenced by:  elrnmpt1s  4681