Theorem elrnmpt1s 4912

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 2193	. . 3 |- C = C
2		elrnmpt1s.1	. . . . 5 |- ( x = D -> B = C )
3	2	eqeq2d 2205	. . . 4 |- ( x = D -> ( C = B <-> C = C ) )
4	3	rspcev 2864	. . 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 4911	. . 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 1364   e. wcel 2164   E.wrex 2473   |-> cmpt 4090   ran crn 4660

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-mpt 4092  df-cnv 4667  df-dm 4669  df-rn 4670

This theorem is referenced by: (None)