Theorem elrnmpt 4928

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 2212	. . 3 |- ( y = C -> ( y = B <-> C = B ) )
2	1	rexbidv 2507	. 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 4927	. 2 |- ran F = { y | E. x e. A y = B }
5	2, 4	elab2g 2920	1 |- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2176   E.wrex 2485   |-> cmpt 4106   ran crn 4677

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-mpt 4108  df-cnv 4684  df-dm 4686  df-rn 4687

This theorem is referenced by:  elrnmpt1s  4929  elrnmptdv  4933  elrnmpt2d  4934  fifo  7084  4sqleminfi  12753  conjnmzb  13649  gausslemma2dlem1a  15568