Theorem elrnmpt1s 4879

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 2177	. . 3 |- C = C
2		elrnmpt1s.1	. . . . 5 |- ( x = D -> B = C )
3	2	eqeq2d 2189	. . . 4 |- ( x = D -> ( C = B <-> C = C ) )
4	3	rspcev 2843	. . 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 4878	. . 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 1353   e. wcel 2148   E.wrex 2456   |-> cmpt 4066   ran crn 4629

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-mpt 4068  df-cnv 4636  df-dm 4638  df-rn 4639

This theorem is referenced by: (None)