Theorem elrnmptdv 4910

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 2871	. 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 4905	. . 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 1364   e. wcel 2164   E.wrex 2473   |-> cmpt 4090   ran crn 4656

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 4663  df-dm 4665  df-rn 4666

This theorem is referenced by: (None)