Theorem ffnafv 47359

Description: A function maps to a class to which all values belong, analogous to ffnfv 7062. (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 6660	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		fafvelcdm 47358	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹'''𝑥) ∈ 𝐵)
3	2	ralrimiva 3126	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵)
4	1, 3	jca 511	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵))
5		simpl 482	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵) → 𝐹 Fn 𝐴)
6		afvelrnb0 47352	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦))
7		nfra1 3258	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵
8		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
9		rsp 3222	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝐹'''𝑥) ∈ 𝐵))
10		eleq1 2822	. . . . . . . 8 ⊢ ((𝐹'''𝑥) = 𝑦 → ((𝐹'''𝑥) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
11	10	biimpcd 249	. . . . . . 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 3241	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵 → (∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦 → 𝑦 ∈ 𝐵))
14	6, 13	sylan9 507	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ 𝐵))
15	14	ssrdv 3937	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵) → ran 𝐹 ⊆ 𝐵)
16		df-f 6494	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
17	5, 15, 16	sylanbrc 583	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵) → 𝐹:𝐴⟶𝐵)
18	4, 17	impbii 209	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058   ⊆ wss 3899  ran crn 5623   Fn wfn 6485  ⟶wf 6486  '''cafv 47305

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-aiota 47273  df-dfat 47307  df-afv 47308

This theorem is referenced by:  ffnaov  47387