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Theorem ffvelrn 5352
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
StepHypRef Expression
1 ffn 5097 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 fnfvelrn 5351 . . 3  |-  ( ( F  Fn  A  /\  C  e.  A )  ->  ( F `  C
)  e.  ran  F
)
31, 2sylan 277 . 2  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( F `  C
)  e.  ran  F
)
4 frn 5103 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
54sseld 3007 . . 3  |-  ( F : A --> B  -> 
( ( F `  C )  e.  ran  F  ->  ( F `  C )  e.  B
) )
65adantr 270 . 2  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( ( F `  C )  e.  ran  F  ->  ( F `  C )  e.  B
) )
73, 6mpd 13 1  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( F `  C
)  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434   ran crn 4392    Fn wfn 4947   -->wf 4948   ` cfv 4952
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fv 4960
This theorem is referenced by:  ffvelrni  5353  ffvelrnda  5354  dffo3  5366  foco2  5370  ffnfv  5375  ffvresb  5380  fcompt  5385  fsn2  5389  fvconst  5403  fcofo  5475  cocan1  5478  isocnv  5502  isores2  5504  isopolem  5512  isosolem  5514  fovrn  5694  off  5775  2dom  6373  enm  6385  xpdom2  6396  isotilem  6513  shftf  9919  nn0seqcvgd  10630  eucialg  10648  phimullem  10808
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