Theorem fnfvelrn 5712

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 5711	. 2 |- ( ( Fun F /\ B e. dom F ) -> ( F ` B ) e. ran F )
2	1	funfni 5376	1 |- ( ( F Fn A /\ B e. A ) -> ( F ` B ) e. ran F )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2176   ran crn 4676   Fn wfn 5266   ` cfv 5271

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279

This theorem is referenced by:  ffvelcdm  5713  fnovrn  6094  fo1stresm  6247  fo2ndresm  6248  fo2ndf  6313  phplem4  6952  phplem4on  6964  cc2lem  7378  frec2uzrand  10550  frecuzrdglem  10556  frecuzrdg0  10558  frecuzrdg0t  10567  ccatrn  11065  uzin2  11298  ghmrn  13593  conjnmz  13615