Theorem elrnmpt2d 4881

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 4875	. . 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 1353   e. wcel 2148   E.wrex 2456   |-> cmpt 4063   ran crn 4626

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-mpt 4065  df-cnv 4633  df-dm 4635  df-rn 4636

This theorem is referenced by: (None)