Theorem elrnmpt2d 4834

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

Hypotheses

Ref	Expression
elrnmpt2d.1	|- F = ( x e. A |-> B )
elrnmpt2d.2	|- ( ph -> C e. ran F )

Assertion

Ref	Expression
elrnmpt2d	|- ( ph -> E. x e. A C = B )

Distinct variable group:   x, C

Allowed substitution hints:   ph( x)   A( x)   B( x)   F( x)

Proof of Theorem elrnmpt2d

Step	Hyp	Ref	Expression
1		elrnmpt2d.2	. 2 |- ( ph -> C e. ran F )
2		elrnmpt2d.1	. . . 4 |- F = ( x e. A |-> B )
3	2	elrnmpt 4828	. . 3 |- ( C e. ran F -> ( C e. ran F <-> E. x e. A C = B ) )
4	3	ibi 175	. 2 |- ( C e. ran F -> E. x e. A C = B )
5	1, 4	syl 14	1 |- ( ph -> E. x e. A C = B )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 2125   E.wrex 2433   |-> cmpt 4021   ran crn 4580

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-opab 4022  df-mpt 4023  df-cnv 4587  df-dm 4589  df-rn 4590

This theorem is referenced by: (None)