Theorem elrnmpt2d 4894

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 4888	. . 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 1363   e. wcel 2158   E.wrex 2466   |-> cmpt 4076   ran crn 4639

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-mpt 4078  df-cnv 4646  df-dm 4648  df-rn 4649

This theorem is referenced by: (None)