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Theorem fafvelrn 28032
Description: A function's value belongs to its codomain, analogous to ffvelrn 5663. (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 5389 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 fnafvelrn 28031 . . 3  |-  ( ( F  Fn  A  /\  C  e.  A )  ->  ( F''' C )  e.  ran  F )
31, 2sylan 457 . 2  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( F''' C )  e.  ran  F )
4 frn 5395 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
54sseld 3179 . . 3  |-  ( F : A --> B  -> 
( ( F''' C )  e.  ran  F  -> 
( F''' C )  e.  B
) )
65adantr 451 . 2  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( ( F''' C )  e.  ran  F  -> 
( F''' C )  e.  B
) )
73, 6mpd 14 1  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( F''' C )  e.  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   ran crn 4690    Fn wfn 5250   -->wf 5251  '''cafv 27972
This theorem is referenced by:  ffnafv  28033
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-dfat 27974  df-afv 27975
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