Theorem elrnmpt1s 4974

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 2229	. . 3 |- C = C
2		elrnmpt1s.1	. . . . 5 |- ( x = D -> B = C )
3	2	eqeq2d 2241	. . . 4 |- ( x = D -> ( C = B <-> C = C ) )
4	3	rspcev 2907	. . 3 |- ( ( D e. A /\ C = C ) -> E. x e. A C = B )
5	1, 4	mpan2 425	. 2 |- ( D e. A -> E. x e. A C = B )
6		rnmpt.1	. . . 4 |- F = ( x e. A |-> B )
7	6	elrnmpt 4973	. . 3 |- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) )
8	7	biimparc 299	. 2 |- ( ( E. x e. A C = B /\ C e. V ) -> C e. ran F )
9	5, 8	sylan 283	1 |- ( ( D e. A /\ C e. V ) -> C e. ran F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   E.wrex 2509   |-> cmpt 4145   ran crn 4720

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 4202  ax-pow 4258  ax-pr 4293

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 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-mpt 4147  df-cnv 4727  df-dm 4729  df-rn 4730

This theorem is referenced by: (None)