Theorem elrnmpt2d 4897

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 4891	. . 3 |- ( C e. ran F -> ( C e. ran F <-> E. x e. A C = B ) )
4	3	ibi 176	. 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 1364   e. wcel 2160   E.wrex 2469   |-> cmpt 4079   ran crn 4642

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 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-mpt 4081  df-cnv 4649  df-dm 4651  df-rn 4652

This theorem is referenced by: (None)