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Theorem fafvelrn 28010
Description: A function's value belongs to its codomain, analogous to ffvelrn 5868. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
fafvelrn  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( F''' C )  e.  B
)

Proof of Theorem fafvelrn
StepHypRef Expression
1 ffn 5591 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 fnafvelrn 28009 . . 3  |-  ( ( F  Fn  A  /\  C  e.  A )  ->  ( F''' C )  e.  ran  F )
31, 2sylan 458 . 2  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( F''' C )  e.  ran  F )
4 frn 5597 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
54sseld 3347 . . 3  |-  ( F : A --> B  -> 
( ( F''' C )  e.  ran  F  -> 
( F''' C )  e.  B
) )
65adantr 452 . 2  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( ( F''' C )  e.  ran  F  -> 
( F''' C )  e.  B
) )
73, 6mpd 15 1  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( F''' C )  e.  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   ran crn 4879    Fn wfn 5449   -->wf 5450  '''cafv 27948
This theorem is referenced by:  ffnafv  28011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-dfat 27950  df-afv 27951
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