Theorem fnfvelrn 5714

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 5713	. 2 |- ( ( Fun F /\ B e. dom F ) -> ( F ` B ) e. ran F )
2	1	funfni 5377	1 |- ( ( F Fn A /\ B e. A ) -> ( F ` B ) e. ran F )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2176   ran crn 4677   Fn wfn 5267   ` cfv 5272

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 4163  ax-pow 4219  ax-pr 4254

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 4046  df-opab 4107  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280

This theorem is referenced by:  ffvelcdm  5715  fnovrn  6096  fo1stresm  6249  fo2ndresm  6250  fo2ndf  6315  phplem4  6954  phplem4on  6966  cc2lem  7380  frec2uzrand  10552  frecuzrdglem  10558  frecuzrdg0  10560  frecuzrdg0t  10569  ccatrn  11068  uzin2  11331  ghmrn  13626  conjnmz  13648