Theorem fnfvelrn 5787

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 5786	. 2 |- ( ( Fun F /\ B e. dom F ) -> ( F ` B ) e. ran F )
2	1	funfni 5439	1 |- ( ( F Fn A /\ B e. A ) -> ( F ` B ) e. ran F )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   ran crn 4732   Fn wfn 5328   ` cfv 5333

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341

This theorem is referenced by:  ffvelcdm  5788  fnovrn  6180  fo1stresm  6333  fo2ndresm  6334  fo2ndf  6401  phplem4  7084  phplem4on  7097  cc2lem  7545  frec2uzrand  10730  frecuzrdglem  10736  frecuzrdg0  10738  frecuzrdg0t  10747  ccatrn  11252  uzin2  11627  ghmrn  13924  conjnmz  13946