Theorem fnfvelrn 5650

Description: A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)

Assertion

Ref	Expression
fnfvelrn	|- ( ( F Fn A /\ B e. A ) -> ( F ` B ) e. ran F )

Proof of Theorem fnfvelrn

Step	Hyp	Ref	Expression
1		fvelrn 5649	. 2 |- ( ( Fun F /\ B e. dom F ) -> ( F ` B ) e. ran F )
2	1	funfni 5318	1 |- ( ( F Fn A /\ B e. A ) -> ( F ` B ) e. ran F )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   ran crn 4629   Fn wfn 5213   ` cfv 5218

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 4123  ax-pow 4176  ax-pr 4211

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-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226

This theorem is referenced by:  ffvelcdm  5651  fnovrn  6024  fo1stresm  6164  fo2ndresm  6165  fo2ndf  6230  phplem4  6857  phplem4on  6869  cc2lem  7267  frec2uzrand  10407  frecuzrdglem  10413  frecuzrdg0  10415  frecuzrdg0t  10424  uzin2  10998