Theorem elrnmptdv 4793

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 2796	. 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 4788	. . 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 1331   e. wcel 1480   E.wrex 2417   |-> cmpt 3989   ran crn 4540

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-mpt 3991  df-cnv 4547  df-dm 4549  df-rn 4550

This theorem is referenced by: (None)