Theorem elrnmpt1s 4602

Description: Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)

Hypotheses

Ref	Expression
rnmpt.1	|- F = ( x e. A |-> B )
elrnmpt1s.1	|- ( x = D -> B = C )

Assertion

Ref	Expression
elrnmpt1s	|- ( ( D e. A /\ C e. V ) -> C e. ran F )

Distinct variable groups:   x, C   x, A   x, D

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

Proof of Theorem elrnmpt1s

Step	Hyp	Ref	Expression
1		eqid 2081	. . 3 |- C = C
2		elrnmpt1s.1	. . . . 5 |- ( x = D -> B = C )
3	2	eqeq2d 2092	. . . 4 |- ( x = D -> ( C = B <-> C = C ) )
4	3	rspcev 2701	. . 3 |- ( ( D e. A /\ C = C ) -> E. x e. A C = B )
5	1, 4	mpan2 415	. 2 |- ( D e. A -> E. x e. A C = B )
6		rnmpt.1	. . . 4 |- F = ( x e. A |-> B )
7	6	elrnmpt 4601	. . 3 |- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) )
8	7	biimparc 293	. 2 |- ( ( E. x e. A C = B /\ C e. V ) -> C e. ran F )
9	5, 8	sylan 277	1 |- ( ( D e. A /\ C e. V ) -> C e. ran F )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   E.wrex 2349   |-> cmpt 3839   ran crn 4364

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-mpt 3841  df-cnv 4371  df-dm 4373  df-rn 4374

This theorem is referenced by: (None)