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Theorem fafvelcdm 45476
Description: A function's value belongs to its codomain, analogous to ffvelcdm 7037. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
fafvelcdm ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹'''𝐶) ∈ 𝐵)

Proof of Theorem fafvelcdm
StepHypRef Expression
1 ffn 6673 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fnafvelrn 45475 . . 3 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹'''𝐶) ∈ ran 𝐹)
31, 2sylan 581 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹'''𝐶) ∈ ran 𝐹)
4 frn 6680 . . . 4 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
54sseld 3948 . . 3 (𝐹:𝐴𝐵 → ((𝐹'''𝐶) ∈ ran 𝐹 → (𝐹'''𝐶) ∈ 𝐵))
65adantr 482 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → ((𝐹'''𝐶) ∈ ran 𝐹 → (𝐹'''𝐶) ∈ 𝐵))
73, 6mpd 15 1 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹'''𝐶) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  ran crn 5639   Fn wfn 6496  wf 6497  '''cafv 45423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-aiota 45391  df-dfat 45425  df-afv 45426
This theorem is referenced by:  ffnafv  45477
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