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Theorem fafvelcdm 47120
Description: A function's value belongs to its codomain, analogous to ffvelcdm 7101. (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 6737 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fnafvelrn 47119 . . 3 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹'''𝐶) ∈ ran 𝐹)
31, 2sylan 580 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹'''𝐶) ∈ ran 𝐹)
4 frn 6744 . . . 4 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
54sseld 3994 . . 3 (𝐹:𝐴𝐵 → ((𝐹'''𝐶) ∈ ran 𝐹 → (𝐹'''𝐶) ∈ 𝐵))
65adantr 480 . 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 395  wcel 2106  ran crn 5690   Fn wfn 6558  wf 6559  '''cafv 47067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-aiota 47035  df-dfat 47069  df-afv 47070
This theorem is referenced by:  ffnafv  47121
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