Theorem elrnmpt 4979

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 2236	. . 3 |- ( y = C -> ( y = B <-> C = B ) )
2	1	rexbidv 2531	. 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 4978	. 2 |- ran F = { y | E. x e. A y = B }
5	2, 4	elab2g 2951	1 |- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   E.wrex 2509   |-> cmpt 4148   ran crn 4724

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-mpt 4150  df-cnv 4731  df-dm 4733  df-rn 4734

This theorem is referenced by:  elrnmpt1s  4980  elrnmptdv  4984  elrnmpt2d  4985  fifo  7170  4sqleminfi  12960  conjnmzb  13857  gausslemma2dlem1a  15777