Theorem ffnafv 43390

Description: A function maps to a class to which all values belong, analogous to ffnfv 6882. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
ffnafv	⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem ffnafv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 6514	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		fafvelrn 43389	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹'''𝑥) ∈ 𝐵)
3	2	ralrimiva 3182	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵)
4	1, 3	jca 514	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵))
5		simpl 485	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵) → 𝐹 Fn 𝐴)
6		afvelrnb0 43383	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦))
7		nfra1 3219	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵
8		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
9		rsp 3205	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝐹'''𝑥) ∈ 𝐵))
10		eleq1 2900	. . . . . . . 8 ⊢ ((𝐹'''𝑥) = 𝑦 → ((𝐹'''𝑥) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
11	10	biimpcd 251	. . . . . . 7 ⊢ ((𝐹'''𝑥) ∈ 𝐵 → ((𝐹'''𝑥) = 𝑦 → 𝑦 ∈ 𝐵))
12	9, 11	syl6 35	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵 → (𝑥 ∈ 𝐴 → ((𝐹'''𝑥) = 𝑦 → 𝑦 ∈ 𝐵)))
13	7, 8, 12	rexlimd 3317	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵 → (∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦 → 𝑦 ∈ 𝐵))
14	6, 13	sylan9 510	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ 𝐵))
15	14	ssrdv 3973	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵) → ran 𝐹 ⊆ 𝐵)
16		df-f 6359	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
17	5, 15, 16	sylanbrc 585	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵) → 𝐹:𝐴⟶𝐵)
18	4, 17	impbii 211	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139   ⊆ wss 3936  ran crn 5556   Fn wfn 6350  ⟶wf 6351  '''cafv 43336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-int 4877  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-aiota 43305  df-dfat 43338  df-afv 43339

This theorem is referenced by:  ffnaov  43418