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Theorem fafvelrn 32421
Description: A function's value belongs to its codomain, analogous to ffvelrn 5931. (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 5639 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 fnafvelrn 32420 . . 3  |-  ( ( F  Fn  A  /\  C  e.  A )  ->  ( F''' C )  e.  ran  F )
31, 2sylan 469 . 2  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( F''' C )  e.  ran  F )
4 frn 5645 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
54sseld 3416 . . 3  |-  ( F : A --> B  -> 
( ( F''' C )  e.  ran  F  -> 
( F''' C )  e.  B
) )
65adantr 463 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1826   ran crn 4914    Fn wfn 5491   -->wf 5492  '''cafv 32365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fv 5504  df-dfat 32367  df-afv 32368
This theorem is referenced by:  ffnafv  32422
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