Theorem fvn0ssdmfun 6952

Description: If a class' function values for certain arguments is not the empty set, the arguments are contained in the domain of the class, and the class restricted to the arguments is a function, analogous to fvfundmfvn0 6812. (Contributed by AV, 27-Jan-2020.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

Assertion

Ref	Expression
fvn0ssdmfun	⊢ (∀𝑎 ∈ 𝐷 (𝐹‘𝑎) ≠ ∅ → (𝐷 ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ 𝐷)))

Distinct variable groups:   𝐷,𝑎   𝐹,𝑎

Proof of Theorem fvn0ssdmfun

Dummy variables 𝑝 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvfundmfvn0 6812	. . 3 ⊢ ((𝐹‘𝑎) ≠ ∅ → (𝑎 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑎})))
2	1	ralimi 3087	. 2 ⊢ (∀𝑎 ∈ 𝐷 (𝐹‘𝑎) ≠ ∅ → ∀𝑎 ∈ 𝐷 (𝑎 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑎})))
3		r19.26 3095	. . 3 ⊢ (∀𝑎 ∈ 𝐷 (𝑎 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑎})) ↔ (∀𝑎 ∈ 𝐷 𝑎 ∈ dom 𝐹 ∧ ∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎})))
4		eleq1w 2821	. . . . . 6 ⊢ (𝑎 = 𝑝 → (𝑎 ∈ dom 𝐹 ↔ 𝑝 ∈ dom 𝐹))
5	4	rspccv 3558	. . . . 5 ⊢ (∀𝑎 ∈ 𝐷 𝑎 ∈ dom 𝐹 → (𝑝 ∈ 𝐷 → 𝑝 ∈ dom 𝐹))
6	5	ssrdv 3927	. . . 4 ⊢ (∀𝑎 ∈ 𝐷 𝑎 ∈ dom 𝐹 → 𝐷 ⊆ dom 𝐹)
7		funrel 6451	. . . . . . . 8 ⊢ (Fun (𝐹 ↾ {𝑎}) → Rel (𝐹 ↾ {𝑎}))
8	7	ralimi 3087	. . . . . . 7 ⊢ (∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎}) → ∀𝑎 ∈ 𝐷 Rel (𝐹 ↾ {𝑎}))
9		reliun 5726	. . . . . . 7 ⊢ (Rel ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) ↔ ∀𝑎 ∈ 𝐷 Rel (𝐹 ↾ {𝑎}))
10	8, 9	sylibr 233	. . . . . 6 ⊢ (∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎}) → Rel ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}))
11		sneq 4571	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑥 → {𝑎} = {𝑥})
12	11	reseq2d 5891	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝐹 ↾ {𝑎}) = (𝐹 ↾ {𝑥}))
13	12	funeqd 6456	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (Fun (𝐹 ↾ {𝑎}) ↔ Fun (𝐹 ↾ {𝑥})))
14	13	rspcva 3559	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐷 ∧ ∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎})) → Fun (𝐹 ↾ {𝑥}))
15		dffun5 6447	. . . . . . . . . . . 12 ⊢ (Fun (𝐹 ↾ {𝑥}) ↔ (Rel (𝐹 ↾ {𝑥}) ∧ ∀𝑤∃𝑦∀𝑧(⟨𝑤, 𝑧⟩ ∈ (𝐹 ↾ {𝑥}) → 𝑧 = 𝑦)))
16		vex 3436	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
17	16	elsnres 5931	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑤, 𝑧⟩ ∈ (𝐹 ↾ {𝑥}) ↔ ∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹))
18	17	imbi1i 350	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑤, 𝑧⟩ ∈ (𝐹 ↾ {𝑥}) → 𝑧 = 𝑦) ↔ (∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹) → 𝑧 = 𝑦))
19	18	albii 1822	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧(⟨𝑤, 𝑧⟩ ∈ (𝐹 ↾ {𝑥}) → 𝑧 = 𝑦) ↔ ∀𝑧(∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹) → 𝑧 = 𝑦))
20	19	exbii 1850	. . . . . . . . . . . . . 14 ⊢ (∃𝑦∀𝑧(⟨𝑤, 𝑧⟩ ∈ (𝐹 ↾ {𝑥}) → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹) → 𝑧 = 𝑦))
21	20	albii 1822	. . . . . . . . . . . . 13 ⊢ (∀𝑤∃𝑦∀𝑧(⟨𝑤, 𝑧⟩ ∈ (𝐹 ↾ {𝑥}) → 𝑧 = 𝑦) ↔ ∀𝑤∃𝑦∀𝑧(∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹) → 𝑧 = 𝑦))
22		equcom 2021	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑧 ↔ 𝑧 = 𝑎)
23		opeq12 4806	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑤 = 𝑥 ∧ 𝑧 = 𝑎) → ⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩)
24	23	ex 413	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑥 → (𝑧 = 𝑎 → ⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩))
25	22, 24	syl5bi 241	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑥 → (𝑎 = 𝑧 → ⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩))
26	25	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 = 𝑥 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → (𝑎 = 𝑧 → ⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩))
27	26	impcom 408	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝑧 ∧ (𝑤 = 𝑥 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹)) → ⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩)
28		opeq2 4805	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = 𝑎 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑎⟩)
29	28	equcoms 2023	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑧 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑎⟩)
30	29	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑧 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑎⟩ ∈ 𝐹))
31	30	biimpcd 248	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝑥, 𝑧⟩ ∈ 𝐹 → (𝑎 = 𝑧 → ⟨𝑥, 𝑎⟩ ∈ 𝐹))
32	31	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 = 𝑥 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → (𝑎 = 𝑧 → ⟨𝑥, 𝑎⟩ ∈ 𝐹))
33	32	impcom 408	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝑧 ∧ (𝑤 = 𝑥 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹)) → ⟨𝑥, 𝑎⟩ ∈ 𝐹)
34	27, 33	jca 512	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = 𝑧 ∧ (𝑤 = 𝑥 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹)) → (⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹))
35	34	ex 413	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑧 → ((𝑤 = 𝑥 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → (⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹)))
36	35	spimevw 1998	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 = 𝑥 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → ∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹))
37	36	ex 413	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑥 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → ∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹)))
38	37	imim1d 82	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑥 → ((∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹) → 𝑧 = 𝑦) → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
39	38	alimdv 1919	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑥 → (∀𝑧(∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹) → 𝑧 = 𝑦) → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
40	39	eximdv 1920	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (∃𝑦∀𝑧(∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹) → 𝑧 = 𝑦) → ∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
41	40	spimvw 1999	. . . . . . . . . . . . 13 ⊢ (∀𝑤∃𝑦∀𝑧(∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹) → 𝑧 = 𝑦) → ∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦))
42	21, 41	sylbi 216	. . . . . . . . . . . 12 ⊢ (∀𝑤∃𝑦∀𝑧(⟨𝑤, 𝑧⟩ ∈ (𝐹 ↾ {𝑥}) → 𝑧 = 𝑦) → ∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦))
43	15, 42	simplbiim 505	. . . . . . . . . . 11 ⊢ (Fun (𝐹 ↾ {𝑥}) → ∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦))
44	14, 43	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐷 ∧ ∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎})) → ∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦))
45	44	expcom 414	. . . . . . . . 9 ⊢ (∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎}) → (𝑥 ∈ 𝐷 → ∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
46		impexp 451	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦) ↔ (𝑥 ∈ 𝐷 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
47	46	albii 1822	. . . . . . . . . . 11 ⊢ (∀𝑧((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦) ↔ ∀𝑧(𝑥 ∈ 𝐷 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
48	47	exbii 1850	. . . . . . . . . 10 ⊢ (∃𝑦∀𝑧((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(𝑥 ∈ 𝐷 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
49		19.21v 1942	. . . . . . . . . . 11 ⊢ (∀𝑧(𝑥 ∈ 𝐷 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)) ↔ (𝑥 ∈ 𝐷 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
50	49	exbii 1850	. . . . . . . . . 10 ⊢ (∃𝑦∀𝑧(𝑥 ∈ 𝐷 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)) ↔ ∃𝑦(𝑥 ∈ 𝐷 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
51		19.37v 1995	. . . . . . . . . 10 ⊢ (∃𝑦(𝑥 ∈ 𝐷 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)) ↔ (𝑥 ∈ 𝐷 → ∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
52	48, 50, 51	3bitri 297	. . . . . . . . 9 ⊢ (∃𝑦∀𝑧((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦) ↔ (𝑥 ∈ 𝐷 → ∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
53	45, 52	sylibr 233	. . . . . . . 8 ⊢ (∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎}) → ∃𝑦∀𝑧((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦))
54	53	alrimiv 1930	. . . . . . 7 ⊢ (∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎}) → ∀𝑥∃𝑦∀𝑧((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦))
55		resiun2 5912	. . . . . . . . . . . . . 14 ⊢ (𝐹 ↾ ∪ 𝑎 ∈ 𝐷 {𝑎}) = ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎})
56	55	eqcomi 2747	. . . . . . . . . . . . 13 ⊢ ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) = (𝐹 ↾ ∪ 𝑎 ∈ 𝐷 {𝑎})
57	56	eleq2i 2830	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑧⟩ ∈ ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) ↔ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ↾ ∪ 𝑎 ∈ 𝐷 {𝑎}))
58		iunid 4990	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑎 ∈ 𝐷 {𝑎} = 𝐷
59	58	reseq2i 5888	. . . . . . . . . . . . 13 ⊢ (𝐹 ↾ ∪ 𝑎 ∈ 𝐷 {𝑎}) = (𝐹 ↾ 𝐷)
60	59	eleq2i 2830	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐹 ↾ ∪ 𝑎 ∈ 𝐷 {𝑎}) ↔ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ↾ 𝐷))
61		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
62	61	opelresi 5899	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐹 ↾ 𝐷) ↔ (𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
63	57, 60, 62	3bitri 297	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑧⟩ ∈ ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) ↔ (𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
64	63	imbi1i 350	. . . . . . . . . 10 ⊢ ((⟨𝑥, 𝑧⟩ ∈ ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) → 𝑧 = 𝑦) ↔ ((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦))
65	64	albii 1822	. . . . . . . . 9 ⊢ (∀𝑧(⟨𝑥, 𝑧⟩ ∈ ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) → 𝑧 = 𝑦) ↔ ∀𝑧((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦))
66	65	exbii 1850	. . . . . . . 8 ⊢ (∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦))
67	66	albii 1822	. . . . . . 7 ⊢ (∀𝑥∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) → 𝑧 = 𝑦) ↔ ∀𝑥∃𝑦∀𝑧((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦))
68	54, 67	sylibr 233	. . . . . 6 ⊢ (∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎}) → ∀𝑥∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) → 𝑧 = 𝑦))
69		dffun5 6447	. . . . . 6 ⊢ (Fun ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) ↔ (Rel ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) ∧ ∀𝑥∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) → 𝑧 = 𝑦)))
70	10, 68, 69	sylanbrc 583	. . . . 5 ⊢ (∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎}) → Fun ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}))
71	58	eqcomi 2747	. . . . . . . 8 ⊢ 𝐷 = ∪ 𝑎 ∈ 𝐷 {𝑎}
72	71	reseq2i 5888	. . . . . . 7 ⊢ (𝐹 ↾ 𝐷) = (𝐹 ↾ ∪ 𝑎 ∈ 𝐷 {𝑎})
73	72	funeqi 6455	. . . . . 6 ⊢ (Fun (𝐹 ↾ 𝐷) ↔ Fun (𝐹 ↾ ∪ 𝑎 ∈ 𝐷 {𝑎}))
74	55	funeqi 6455	. . . . . 6 ⊢ (Fun (𝐹 ↾ ∪ 𝑎 ∈ 𝐷 {𝑎}) ↔ Fun ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}))
75	73, 74	bitri 274	. . . . 5 ⊢ (Fun (𝐹 ↾ 𝐷) ↔ Fun ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}))
76	70, 75	sylibr 233	. . . 4 ⊢ (∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎}) → Fun (𝐹 ↾ 𝐷))
77	6, 76	anim12i 613	. . 3 ⊢ ((∀𝑎 ∈ 𝐷 𝑎 ∈ dom 𝐹 ∧ ∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎})) → (𝐷 ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ 𝐷)))
78	3, 77	sylbi 216	. 2 ⊢ (∀𝑎 ∈ 𝐷 (𝑎 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑎})) → (𝐷 ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ 𝐷)))
79	2, 78	syl 17	1 ⊢ (∀𝑎 ∈ 𝐷 (𝐹‘𝑎) ≠ ∅ → (𝐷 ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064   ⊆ wss 3887  ∅c0 4256  {csn 4561  ⟨cop 4567  ∪ ciun 4924  dom cdm 5589   ↾ cres 5591  Rel wrel 5594  Fun wfun 6427  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6391  df-fun 6435  df-fv 6441

This theorem is referenced by:  fveqressseq  6957  ovn0ssdmfun  45321