Theorem elrnmptdv 4954

Description: Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)

Hypotheses

Ref	Expression
elrnmptdv.1	|- F = ( x e. A |-> B )
elrnmptdv.2	|- ( ph -> C e. A )
elrnmptdv.3	|- ( ph -> D e. V )
elrnmptdv.4	|- ( ( ph /\ x = C ) -> D = B )

Assertion

Ref	Expression
elrnmptdv	|- ( ph -> D e. ran F )

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

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

Proof of Theorem elrnmptdv

Step	Hyp	Ref	Expression
1		elrnmptdv.4	. . 3 |- ( ( ph /\ x = C ) -> D = B )
2		elrnmptdv.2	. . 3 |- ( ph -> C e. A )
3	1, 2	rspcime 2894	. 2 |- ( ph -> E. x e. A D = B )
4		elrnmptdv.3	. . 3 |- ( ph -> D e. V )
5		elrnmptdv.1	. . . 4 |- F = ( x e. A |-> B )
6	5	elrnmpt 4949	. . 3 |- ( D e. V -> ( D e. ran F <-> E. x e. A D = B ) )
7	4, 6	syl 14	. 2 |- ( ph -> ( D e. ran F <-> E. x e. A D = B ) )
8	3, 7	mpbird 167	1 |- ( ph -> D e. ran F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1375   e. wcel 2180   E.wrex 2489   |-> cmpt 4124   ran crn 4697

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-rex 2494  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-br 4063  df-opab 4125  df-mpt 4126  df-cnv 4704  df-dm 4706  df-rn 4707

This theorem is referenced by: (None)