Theorem fafvelcdm 47166

Description: A function's value belongs to its codomain, analogous to ffvelcdm 7076. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
fafvelcdm	⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹'''𝐶) ∈ 𝐵)

Proof of Theorem fafvelcdm

Step	Hyp	Ref	Expression
1		ffn 6711	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		fnafvelrn 47165	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹'''𝐶) ∈ ran 𝐹)
3	1, 2	sylan 580	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹'''𝐶) ∈ ran 𝐹)
4		frn 6718	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
5	4	sseld 3962	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹'''𝐶) ∈ ran 𝐹 → (𝐹'''𝐶) ∈ 𝐵))
6	5	adantr 480	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹'''𝐶) ∈ ran 𝐹 → (𝐹'''𝐶) ∈ 𝐵))
7	3, 6	mpd 15	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹'''𝐶) ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ran crn 5660   Fn wfn 6531  ⟶wf 6532  '''cafv 47113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-aiota 47081  df-dfat 47115  df-afv 47116

This theorem is referenced by:  ffnafv  47167