Theorem fvn0ssdmfun 7085

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 6936. (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 6936	. . 3 ⊢ ((𝐹‘𝑎) ≠ ∅ → (𝑎 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑎})))
2	1	ralimi 3103	. 2 ⊢ (∀𝑎 ∈ 𝐷 (𝐹‘𝑎) ≠ ∅ → ∀𝑎 ∈ 𝐷 (𝑎 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑎})))
3		r19.26 3126	. . 3 ⊢ (∀𝑎 ∈ 𝐷 (𝑎 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑎})) ↔ (∀𝑎 ∈ 𝐷 𝑎 ∈ dom 𝐹 ∧ ∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎})))
4		eleq1w 2847	. . . . . 6 ⊢ (𝑎 = 𝑝 → (𝑎 ∈ dom 𝐹 ↔ 𝑝 ∈ dom 𝐹))
5	4	rspccv 3577	. . . . 5 ⊢ (∀𝑎 ∈ 𝐷 𝑎 ∈ dom 𝐹 → (𝑝 ∈ 𝐷 → 𝑝 ∈ dom 𝐹))
6	5	ssrdv 3919	. . . 4 ⊢ (∀𝑎 ∈ 𝐷 𝑎 ∈ dom 𝐹 → 𝐷 ⊆ dom 𝐹)
7		funrel 6565	. . . . . . . 8 ⊢ (Fun (𝐹 ↾ {𝑎}) → Rel (𝐹 ↾ {𝑎}))
8	7	ralimi 3103	. . . . . . 7 ⊢ (∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎}) → ∀𝑎 ∈ 𝐷 Rel (𝐹 ↾ {𝑎}))
9		reliun 5799	. . . . . . 7 ⊢ (Rel ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) ↔ ∀𝑎 ∈ 𝐷 Rel (𝐹 ↾ {𝑎}))
10	8, 9	sylibr 237	. . . . . 6 ⊢ (∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎}) → Rel ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}))
11		sneq 4594	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑥 → {𝑎} = {𝑥})
12	11	reseq2d 5975	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝐹 ↾ {𝑎}) = (𝐹 ↾ {𝑥}))
13	12	funeqd 6570	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (Fun (𝐹 ↾ {𝑎}) ↔ Fun (𝐹 ↾ {𝑥})))
14	13	rspcva 3578	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐷 ∧ ∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎})) → Fun (𝐹 ↾ {𝑥}))
15		dffun5 6561	. . . . . . . . . . . 12 ⊢ (Fun (𝐹 ↾ {𝑥}) ↔ (Rel (𝐹 ↾ {𝑥}) ∧ ∀𝑤∃𝑦∀𝑧(⟨𝑤, 𝑧⟩ ∈ (𝐹 ↾ {𝑥}) → 𝑧 = 𝑦)))
16		vex 3458	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
17	16	elsnres 6016	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑤, 𝑧⟩ ∈ (𝐹 ↾ {𝑥}) ↔ ∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹))
18	17	imbi1i 352	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑤, 𝑧⟩ ∈ (𝐹 ↾ {𝑥}) → 𝑧 = 𝑦) ↔ (∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹) → 𝑧 = 𝑦))
19	18	albii 1851	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧(⟨𝑤, 𝑧⟩ ∈ (𝐹 ↾ {𝑥}) → 𝑧 = 𝑦) ↔ ∀𝑧(∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹) → 𝑧 = 𝑦))
20	19	exbii 1880	. . . . . . . . . . . . . 14 ⊢ (∃𝑦∀𝑧(⟨𝑤, 𝑧⟩ ∈ (𝐹 ↾ {𝑥}) → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹) → 𝑧 = 𝑦))
21	20	albii 1851	. . . . . . . . . . . . 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 2052	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑧 ↔ 𝑧 = 𝑎)
23		opeq12 4845	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑤 = 𝑥 ∧ 𝑧 = 𝑎) → ⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩)
24	23	ex 418	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑥 → (𝑧 = 𝑎 → ⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩))
25	22, 24	biimtrid 245	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑥 → (𝑎 = 𝑧 → ⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩))
26	25	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 = 𝑥 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → (𝑎 = 𝑧 → ⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩))
27	26	impcom 413	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝑧 ∧ (𝑤 = 𝑥 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹)) → ⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩)
28		opeq2 4844	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = 𝑎 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑎⟩)
29	28	equcoms 2054	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑧 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑎⟩)
30	29	eleq1d 2849	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑧 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑎⟩ ∈ 𝐹))
31	30	biimpcd 252	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝑥, 𝑧⟩ ∈ 𝐹 → (𝑎 = 𝑧 → ⟨𝑥, 𝑎⟩ ∈ 𝐹))
32	31	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 = 𝑥 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → (𝑎 = 𝑧 → ⟨𝑥, 𝑎⟩ ∈ 𝐹))
33	32	impcom 413	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝑧 ∧ (𝑤 = 𝑥 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹)) → ⟨𝑥, 𝑎⟩ ∈ 𝐹)
34	27, 33	jca 521	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = 𝑧 ∧ (𝑤 = 𝑥 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹)) → (⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹))
35	34	ex 418	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑧 → ((𝑤 = 𝑥 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → (⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹)))
36	35	spimevw 2019	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 = 𝑥 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → ∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹))
37	36	ex 418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑥 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → ∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹)))
38	37	imim1d 83	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑥 → ((∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹) → 𝑧 = 𝑦) → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
39	38	alimdv 1950	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑥 → (∀𝑧(∃𝑎(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑎⟩ ∧ ⟨𝑥, 𝑎⟩ ∈ 𝐹) → 𝑧 = 𝑦) → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
40	39	eximdv 1951	. . . . . . . . . . . . . 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 2020	. . . . . . . . . . . . 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 220	. . . . . . . . . . . 12 ⊢ (∀𝑤∃𝑦∀𝑧(⟨𝑤, 𝑧⟩ ∈ (𝐹 ↾ {𝑥}) → 𝑧 = 𝑦) → ∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦))
43	15, 42	simplbiim 514	. . . . . . . . . . 11 ⊢ (Fun (𝐹 ↾ {𝑥}) → ∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦))
44	14, 43	syl 18	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐷 ∧ ∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎})) → ∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦))
45	44	expcom 419	. . . . . . . . 9 ⊢ (∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎}) → (𝑥 ∈ 𝐷 → ∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
46		impexp 456	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦) ↔ (𝑥 ∈ 𝐷 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
47	46	albii 1851	. . . . . . . . . . 11 ⊢ (∀𝑧((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦) ↔ ∀𝑧(𝑥 ∈ 𝐷 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
48	47	exbii 1880	. . . . . . . . . 10 ⊢ (∃𝑦∀𝑧((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(𝑥 ∈ 𝐷 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
49		19.21v 1973	. . . . . . . . . . 11 ⊢ (∀𝑧(𝑥 ∈ 𝐷 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)) ↔ (𝑥 ∈ 𝐷 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
50	49	exbii 1880	. . . . . . . . . 10 ⊢ (∃𝑦∀𝑧(𝑥 ∈ 𝐷 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)) ↔ ∃𝑦(𝑥 ∈ 𝐷 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
51		19.37v 2031	. . . . . . . . . 10 ⊢ (∃𝑦(𝑥 ∈ 𝐷 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)) ↔ (𝑥 ∈ 𝐷 → ∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
52	48, 50, 51	3bitri 300	. . . . . . . . 9 ⊢ (∃𝑦∀𝑧((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦) ↔ (𝑥 ∈ 𝐷 → ∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑧 = 𝑦)))
53	45, 52	sylibr 237	. . . . . . . 8 ⊢ (∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎}) → ∃𝑦∀𝑧((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦))
54	53	alrimiv 1961	. . . . . . 7 ⊢ (∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎}) → ∀𝑥∃𝑦∀𝑧((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦))
55		resiun2 5996	. . . . . . . . . . . . . 14 ⊢ (𝐹 ↾ ∪ 𝑎 ∈ 𝐷 {𝑎}) = ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎})
56	55	eqcomi 2772	. . . . . . . . . . . . 13 ⊢ ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) = (𝐹 ↾ ∪ 𝑎 ∈ 𝐷 {𝑎})
57	56	eleq2i 2856	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑧⟩ ∈ ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) ↔ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ↾ ∪ 𝑎 ∈ 𝐷 {𝑎}))
58		iunid 5022	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑎 ∈ 𝐷 {𝑎} = 𝐷
59	58	reseq2i 5972	. . . . . . . . . . . . 13 ⊢ (𝐹 ↾ ∪ 𝑎 ∈ 𝐷 {𝑎}) = (𝐹 ↾ 𝐷)
60	59	eleq2i 2856	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐹 ↾ ∪ 𝑎 ∈ 𝐷 {𝑎}) ↔ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ↾ 𝐷))
61		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
62	61	opelresi 5983	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐹 ↾ 𝐷) ↔ (𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
63	57, 60, 62	3bitri 300	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑧⟩ ∈ ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) ↔ (𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
64	63	imbi1i 352	. . . . . . . . . 10 ⊢ ((⟨𝑥, 𝑧⟩ ∈ ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) → 𝑧 = 𝑦) ↔ ((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦))
65	64	albii 1851	. . . . . . . . 9 ⊢ (∀𝑧(⟨𝑥, 𝑧⟩ ∈ ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) → 𝑧 = 𝑦) ↔ ∀𝑧((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦))
66	65	exbii 1880	. . . . . . . 8 ⊢ (∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧((𝑥 ∈ 𝐷 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑧 = 𝑦))
67	66	albii 1851	. . . . . . 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 237	. . . . . 6 ⊢ (∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎}) → ∀𝑥∃𝑦∀𝑧(⟨𝑥, 𝑧⟩ ∈ ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}) → 𝑧 = 𝑦))
69		dffun5 6561	. . . . . 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 595	. . . . 5 ⊢ (∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎}) → Fun ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}))
71	58	eqcomi 2772	. . . . . . . 8 ⊢ 𝐷 = ∪ 𝑎 ∈ 𝐷 {𝑎}
72	71	reseq2i 5972	. . . . . . 7 ⊢ (𝐹 ↾ 𝐷) = (𝐹 ↾ ∪ 𝑎 ∈ 𝐷 {𝑎})
73	72	funeqi 6569	. . . . . 6 ⊢ (Fun (𝐹 ↾ 𝐷) ↔ Fun (𝐹 ↾ ∪ 𝑎 ∈ 𝐷 {𝑎}))
74	55	funeqi 6569	. . . . . 6 ⊢ (Fun (𝐹 ↾ ∪ 𝑎 ∈ 𝐷 {𝑎}) ↔ Fun ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}))
75	73, 74	bitri 278	. . . . 5 ⊢ (Fun (𝐹 ↾ 𝐷) ↔ Fun ∪ 𝑎 ∈ 𝐷 (𝐹 ↾ {𝑎}))
76	70, 75	sylibr 237	. . . 4 ⊢ (∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎}) → Fun (𝐹 ↾ 𝐷))
77	6, 76	anim12i 625	. . 3 ⊢ ((∀𝑎 ∈ 𝐷 𝑎 ∈ dom 𝐹 ∧ ∀𝑎 ∈ 𝐷 Fun (𝐹 ↾ {𝑎})) → (𝐷 ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ 𝐷)))
78	3, 77	sylbi 220	. 2 ⊢ (∀𝑎 ∈ 𝐷 (𝑎 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑎})) → (𝐷 ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ 𝐷)))
79	2, 78	syl 18	1 ⊢ (∀𝑎 ∈ 𝐷 (𝐹‘𝑎) ≠ ∅ → (𝐷 ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ 𝐷)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2146   ≠ wne 2959  ∀wral 3080   ⊆ wss 3900  ∅c0 4281  {csn 4588  ⟨cop 4838  ∪ ciun 4954  dom cdm 5655   ↾ cres 5657  Rel wrel 5660  Fun wfun 6541  ‘cfv 6547

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-11 2196  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-res 5667  df-iota 6501  df-fun 6549  df-fv 6555

This theorem is used by:  fveqressseq  7090  ovn0ssdmfun  7599