Theorem ffvelrn 5321

Description: A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)

Assertion

Ref	Expression
ffvelrn	|- ( ( F : A --> B /\ C e. A ) -> ( F ` C ) e. B )

Proof of Theorem ffvelrn

Step	Hyp	Ref	Expression
1		ffn 5066	. . 3 |- ( F : A --> B -> F Fn A )
2		fnfvelrn 5320	. . 3 |- ( ( F Fn A /\ C e. A ) -> ( F ` C ) e. ran F )
3	1, 2	sylan 277	. 2 |- ( ( F : A --> B /\ C e. A ) -> ( F ` C ) e. ran F )
4		frn 5072	. . . 4 |- ( F : A --> B -> ran F C_ B )
5	4	sseld 2998	. . 3 |- ( F : A --> B -> ( ( F ` C ) e. ran F -> ( F ` C ) e. B ) )
6	5	adantr 270	. 2 |- ( ( F : A --> B /\ C e. A ) -> ( ( F ` C ) e. ran F -> ( F ` C ) e. B ) )
7	3, 6	mpd 13	1 |- ( ( F : A --> B /\ C e. A ) -> ( F ` C ) e. B )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1433   ran crn 4364   Fn wfn 4917   -->wf 4918   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930

This theorem is referenced by:  ffvelrni  5322  ffvelrnda  5323  dffo3  5335  foco2  5339  ffnfv  5344  ffvresb  5349  fcompt  5354  fsn2  5358  fvconst  5372  fcofo  5444  cocan1  5447  isocnv  5471  isores2  5473  isopolem  5481  isosolem  5483  fovrn  5663  off  5744  2dom  6308  enm  6317  xpdom2  6328  isotilem  6419  shftf  9718  nn0seqcvgd  10423  eucialg  10441