Theorem elrnmptdv 4858

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 2837	. 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 4853	. . 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 166	1 |- ( ph -> D e. ran F )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   E.wrex 2445   |-> cmpt 4043   ran crn 4605

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-mpt 4045  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by: (None)