Theorem elrnmpt1s 4861

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 2170	. . 3 |- C = C
2		elrnmpt1s.1	. . . . 5 |- ( x = D -> B = C )
3	2	eqeq2d 2182	. . . 4 |- ( x = D -> ( C = B <-> C = C ) )
4	3	rspcev 2834	. . 3 |- ( ( D e. A /\ C = C ) -> E. x e. A C = B )
5	1, 4	mpan2 423	. 2 |- ( D e. A -> E. x e. A C = B )
6		rnmpt.1	. . . 4 |- F = ( x e. A |-> B )
7	6	elrnmpt 4860	. . 3 |- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) )
8	7	biimparc 297	. 2 |- ( ( E. x e. A C = B /\ C e. V ) -> C e. ran F )
9	5, 8	sylan 281	1 |- ( ( D e. A /\ C e. V ) -> C e. ran F )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   E.wrex 2449   |-> cmpt 4050   ran crn 4612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-mpt 4052  df-cnv 4619  df-dm 4621  df-rn 4622

This theorem is referenced by: (None)