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Theorem fafvelrn 30225
Description: A function's value belongs to its codomain, analogous to ffvelrn 5951. (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 5668 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 fnafvelrn 30224 . . 3  |-  ( ( F  Fn  A  /\  C  e.  A )  ->  ( F''' C )  e.  ran  F )
31, 2sylan 471 . 2  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( F''' C )  e.  ran  F )
4 frn 5674 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
54sseld 3464 . . 3  |-  ( F : A --> B  -> 
( ( F''' C )  e.  ran  F  -> 
( F''' C )  e.  B
) )
65adantr 465 . 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 369    e. wcel 1758   ran crn 4950    Fn wfn 5522   -->wf 5523  '''cafv 30167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-dfat 30169  df-afv 30170
This theorem is referenced by:  ffnafv  30226
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