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Theorem fafvelrn 41571
 Description: A function's value belongs to its codomain, analogous to ffvelrn 6397. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
fafvelrn ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹'''𝐶) ∈ 𝐵)

Proof of Theorem fafvelrn
StepHypRef Expression
1 ffn 6083 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fnafvelrn 41570 . . 3 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹'''𝐶) ∈ ran 𝐹)
31, 2sylan 487 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹'''𝐶) ∈ ran 𝐹)
4 frn 6091 . . . 4 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
54sseld 3635 . . 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 383   ∈ wcel 2030  ran crn 5144   Fn wfn 5921  ⟶wf 5922  '''cafv 41515 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-dfat 41517  df-afv 41518 This theorem is referenced by:  ffnafv  41572
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