Theorem fnfvelrn 5843

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 5842	. 2 |- ( ( Fun F /\ B e. dom F ) -> ( F ` B ) e. ran F )
2	1	funfni 5486	1 |- ( ( F Fn A /\ B e. A ) -> ( F ` B ) e. ran F )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   ran crn 4777   Fn wfn 5375   ` cfv 5380

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-iota 5340  df-fun 5382  df-fn 5383  df-fv 5388

This theorem is used by:  ffvelcdm  5844  fnovrn  6240  fo1stresm  6399  fo2ndresm  6400  fo2ndf  6467  phplem4  7160  phplem4on  7173  cc2lem  7638  frec2uzrand  10863  frecuzrdglem  10869  frecuzrdg0  10871  frecuzrdg0t  10880  ccatrn  11399  uzin2  11775  ghmrn  14119  conjnmz  14141  issubassa2  15144