Theorem permac8prim 45291

Description: The Axiom of Choice ac8prim 45268 holds in permutation models. Part of Exercise II.9.3 of [Kunen2] p. 149. Note that ax-ac 10373 requires Regularity for its derivation from the usual Axiom of Choice and does not necessarily hold in permutation models. (Contributed by Eric Schmidt, 16-Nov-2025.)

Hypotheses

Ref	Expression
permmodel.1	⊢ 𝐹:V–1-1-onto→V
permmodel.2	⊢ 𝑅 = (◡𝐹 ∘ E )

Assertion

Ref	Expression
permac8prim	⊢ ((∀𝑧(𝑧𝑅𝑥 → ∃𝑤 𝑤𝑅𝑧) ∧ ∀𝑧∀𝑤((𝑧𝑅𝑥 ∧ 𝑤𝑅𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦𝑅𝑧 → ¬ 𝑦𝑅𝑤)))) → ∃𝑦∀𝑧(𝑧𝑅𝑥 → ∃𝑤∀𝑣((𝑣𝑅𝑧 ∧ 𝑣𝑅𝑦) ↔ 𝑣 = 𝑤)))

Distinct variable groups:   𝑥,𝑧,𝑦,𝑤,𝑣   𝑦,𝐹,𝑧,𝑤,𝑣

Allowed substitution hints:   𝑅(𝑥,𝑦,𝑧,𝑤,𝑣)   𝐹(𝑥)

Proof of Theorem permac8prim

Dummy variables 𝑞 𝑟 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ral 3053	. . . 4 ⊢ (∀𝑧 ∈ (𝐹‘𝑥)(𝐹‘𝑧) ≠ ∅ ↔ ∀𝑧(𝑧 ∈ (𝐹‘𝑥) → (𝐹‘𝑧) ≠ ∅))
2		permmodel.1	. . . . . 6 ⊢ 𝐹:V–1-1-onto→V
3		f1ofn 6776	. . . . . 6 ⊢ (𝐹:V–1-1-onto→V → 𝐹 Fn V)
4	2, 3	ax-mp 5	. . . . 5 ⊢ 𝐹 Fn V
5		ssv 3959	. . . . 5 ⊢ (𝐹‘𝑥) ⊆ V
6		neeq1 2995	. . . . . 6 ⊢ (𝑡 = (𝐹‘𝑧) → (𝑡 ≠ ∅ ↔ (𝐹‘𝑧) ≠ ∅))
7	6	ralima 7185	. . . . 5 ⊢ ((𝐹 Fn V ∧ (𝐹‘𝑥) ⊆ V) → (∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))𝑡 ≠ ∅ ↔ ∀𝑧 ∈ (𝐹‘𝑥)(𝐹‘𝑧) ≠ ∅))
8	4, 5, 7	mp2an 693	. . . 4 ⊢ (∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))𝑡 ≠ ∅ ↔ ∀𝑧 ∈ (𝐹‘𝑥)(𝐹‘𝑧) ≠ ∅)
9		permmodel.2	. . . . . . 7 ⊢ 𝑅 = (◡𝐹 ∘ E )
10		vex 3445	. . . . . . 7 ⊢ 𝑧 ∈ V
11		vex 3445	. . . . . . 7 ⊢ 𝑥 ∈ V
12	2, 9, 10, 11	brpermmodel 45280	. . . . . 6 ⊢ (𝑧𝑅𝑥 ↔ 𝑧 ∈ (𝐹‘𝑥))
13		vex 3445	. . . . . . . . 9 ⊢ 𝑤 ∈ V
14	2, 9, 13, 10	brpermmodel 45280	. . . . . . . 8 ⊢ (𝑤𝑅𝑧 ↔ 𝑤 ∈ (𝐹‘𝑧))
15	14	exbii 1850	. . . . . . 7 ⊢ (∃𝑤 𝑤𝑅𝑧 ↔ ∃𝑤 𝑤 ∈ (𝐹‘𝑧))
16		n0 4306	. . . . . . 7 ⊢ ((𝐹‘𝑧) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝐹‘𝑧))
17	15, 16	bitr4i 278	. . . . . 6 ⊢ (∃𝑤 𝑤𝑅𝑧 ↔ (𝐹‘𝑧) ≠ ∅)
18	12, 17	imbi12i 350	. . . . 5 ⊢ ((𝑧𝑅𝑥 → ∃𝑤 𝑤𝑅𝑧) ↔ (𝑧 ∈ (𝐹‘𝑥) → (𝐹‘𝑧) ≠ ∅))
19	18	albii 1821	. . . 4 ⊢ (∀𝑧(𝑧𝑅𝑥 → ∃𝑤 𝑤𝑅𝑧) ↔ ∀𝑧(𝑧 ∈ (𝐹‘𝑥) → (𝐹‘𝑧) ≠ ∅))
20	1, 8, 19	3bitr4i 303	. . 3 ⊢ (∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))𝑡 ≠ ∅ ↔ ∀𝑧(𝑧𝑅𝑥 → ∃𝑤 𝑤𝑅𝑧))
21		neeq2 2996	. . . . . . . . 9 ⊢ (𝑞 = (𝐹‘𝑤) → (𝑡 ≠ 𝑞 ↔ 𝑡 ≠ (𝐹‘𝑤)))
22		ineq2 4167	. . . . . . . . . 10 ⊢ (𝑞 = (𝐹‘𝑤) → (𝑡 ∩ 𝑞) = (𝑡 ∩ (𝐹‘𝑤)))
23	22	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑞 = (𝐹‘𝑤) → ((𝑡 ∩ 𝑞) = ∅ ↔ (𝑡 ∩ (𝐹‘𝑤)) = ∅))
24	21, 23	imbi12d 344	. . . . . . . 8 ⊢ (𝑞 = (𝐹‘𝑤) → ((𝑡 ≠ 𝑞 → (𝑡 ∩ 𝑞) = ∅) ↔ (𝑡 ≠ (𝐹‘𝑤) → (𝑡 ∩ (𝐹‘𝑤)) = ∅)))
25	24	ralima 7185	. . . . . . 7 ⊢ ((𝐹 Fn V ∧ (𝐹‘𝑥) ⊆ V) → (∀𝑞 ∈ (𝐹 “ (𝐹‘𝑥))(𝑡 ≠ 𝑞 → (𝑡 ∩ 𝑞) = ∅) ↔ ∀𝑤 ∈ (𝐹‘𝑥)(𝑡 ≠ (𝐹‘𝑤) → (𝑡 ∩ (𝐹‘𝑤)) = ∅)))
26	4, 5, 25	mp2an 693	. . . . . 6 ⊢ (∀𝑞 ∈ (𝐹 “ (𝐹‘𝑥))(𝑡 ≠ 𝑞 → (𝑡 ∩ 𝑞) = ∅) ↔ ∀𝑤 ∈ (𝐹‘𝑥)(𝑡 ≠ (𝐹‘𝑤) → (𝑡 ∩ (𝐹‘𝑤)) = ∅))
27	26	ralbii 3083	. . . . 5 ⊢ (∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∀𝑞 ∈ (𝐹 “ (𝐹‘𝑥))(𝑡 ≠ 𝑞 → (𝑡 ∩ 𝑞) = ∅) ↔ ∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∀𝑤 ∈ (𝐹‘𝑥)(𝑡 ≠ (𝐹‘𝑤) → (𝑡 ∩ (𝐹‘𝑤)) = ∅))
28		neeq1 2995	. . . . . . . . 9 ⊢ (𝑡 = (𝐹‘𝑧) → (𝑡 ≠ (𝐹‘𝑤) ↔ (𝐹‘𝑧) ≠ (𝐹‘𝑤)))
29		ineq1 4166	. . . . . . . . . 10 ⊢ (𝑡 = (𝐹‘𝑧) → (𝑡 ∩ (𝐹‘𝑤)) = ((𝐹‘𝑧) ∩ (𝐹‘𝑤)))
30	29	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑡 = (𝐹‘𝑧) → ((𝑡 ∩ (𝐹‘𝑤)) = ∅ ↔ ((𝐹‘𝑧) ∩ (𝐹‘𝑤)) = ∅))
31	28, 30	imbi12d 344	. . . . . . . 8 ⊢ (𝑡 = (𝐹‘𝑧) → ((𝑡 ≠ (𝐹‘𝑤) → (𝑡 ∩ (𝐹‘𝑤)) = ∅) ↔ ((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ((𝐹‘𝑧) ∩ (𝐹‘𝑤)) = ∅)))
32	31	ralbidv 3160	. . . . . . 7 ⊢ (𝑡 = (𝐹‘𝑧) → (∀𝑤 ∈ (𝐹‘𝑥)(𝑡 ≠ (𝐹‘𝑤) → (𝑡 ∩ (𝐹‘𝑤)) = ∅) ↔ ∀𝑤 ∈ (𝐹‘𝑥)((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ((𝐹‘𝑧) ∩ (𝐹‘𝑤)) = ∅)))
33	32	ralima 7185	. . . . . 6 ⊢ ((𝐹 Fn V ∧ (𝐹‘𝑥) ⊆ V) → (∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∀𝑤 ∈ (𝐹‘𝑥)(𝑡 ≠ (𝐹‘𝑤) → (𝑡 ∩ (𝐹‘𝑤)) = ∅) ↔ ∀𝑧 ∈ (𝐹‘𝑥)∀𝑤 ∈ (𝐹‘𝑥)((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ((𝐹‘𝑧) ∩ (𝐹‘𝑤)) = ∅)))
34	4, 5, 33	mp2an 693	. . . . 5 ⊢ (∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∀𝑤 ∈ (𝐹‘𝑥)(𝑡 ≠ (𝐹‘𝑤) → (𝑡 ∩ (𝐹‘𝑤)) = ∅) ↔ ∀𝑧 ∈ (𝐹‘𝑥)∀𝑤 ∈ (𝐹‘𝑥)((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ((𝐹‘𝑧) ∩ (𝐹‘𝑤)) = ∅))
35		r2al 3173	. . . . 5 ⊢ (∀𝑧 ∈ (𝐹‘𝑥)∀𝑤 ∈ (𝐹‘𝑥)((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ((𝐹‘𝑧) ∩ (𝐹‘𝑤)) = ∅) ↔ ∀𝑧∀𝑤((𝑧 ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ (𝐹‘𝑥)) → ((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ((𝐹‘𝑧) ∩ (𝐹‘𝑤)) = ∅)))
36	27, 34, 35	3bitri 297	. . . 4 ⊢ (∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∀𝑞 ∈ (𝐹 “ (𝐹‘𝑥))(𝑡 ≠ 𝑞 → (𝑡 ∩ 𝑞) = ∅) ↔ ∀𝑧∀𝑤((𝑧 ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ (𝐹‘𝑥)) → ((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ((𝐹‘𝑧) ∩ (𝐹‘𝑤)) = ∅)))
37	2, 9, 13, 11	brpermmodel 45280	. . . . . . 7 ⊢ (𝑤𝑅𝑥 ↔ 𝑤 ∈ (𝐹‘𝑥))
38	12, 37	anbi12i 629	. . . . . 6 ⊢ ((𝑧𝑅𝑥 ∧ 𝑤𝑅𝑥) ↔ (𝑧 ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ (𝐹‘𝑥)))
39		df-ne 2934	. . . . . . . 8 ⊢ ((𝐹‘𝑧) ≠ (𝐹‘𝑤) ↔ ¬ (𝐹‘𝑧) = (𝐹‘𝑤))
40		f1of1 6774	. . . . . . . . . . . 12 ⊢ (𝐹:V–1-1-onto→V → 𝐹:V–1-1→V)
41	2, 40	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝐹:V–1-1→V
42		f1fveq 7210	. . . . . . . . . . 11 ⊢ ((𝐹:V–1-1→V ∧ (𝑧 ∈ V ∧ 𝑤 ∈ V)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤))
43	41, 42	mpan 691	. . . . . . . . . 10 ⊢ ((𝑧 ∈ V ∧ 𝑤 ∈ V) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤))
44	43	el2v 3448	. . . . . . . . 9 ⊢ ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤)
45	44	notbii 320	. . . . . . . 8 ⊢ (¬ (𝐹‘𝑧) = (𝐹‘𝑤) ↔ ¬ 𝑧 = 𝑤)
46	39, 45	bitr2i 276	. . . . . . 7 ⊢ (¬ 𝑧 = 𝑤 ↔ (𝐹‘𝑧) ≠ (𝐹‘𝑤))
47		vex 3445	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
48	2, 9, 47, 10	brpermmodel 45280	. . . . . . . . . 10 ⊢ (𝑦𝑅𝑧 ↔ 𝑦 ∈ (𝐹‘𝑧))
49	2, 9, 47, 13	brpermmodel 45280	. . . . . . . . . . 11 ⊢ (𝑦𝑅𝑤 ↔ 𝑦 ∈ (𝐹‘𝑤))
50	49	notbii 320	. . . . . . . . . 10 ⊢ (¬ 𝑦𝑅𝑤 ↔ ¬ 𝑦 ∈ (𝐹‘𝑤))
51	48, 50	imbi12i 350	. . . . . . . . 9 ⊢ ((𝑦𝑅𝑧 → ¬ 𝑦𝑅𝑤) ↔ (𝑦 ∈ (𝐹‘𝑧) → ¬ 𝑦 ∈ (𝐹‘𝑤)))
52	51	albii 1821	. . . . . . . 8 ⊢ (∀𝑦(𝑦𝑅𝑧 → ¬ 𝑦𝑅𝑤) ↔ ∀𝑦(𝑦 ∈ (𝐹‘𝑧) → ¬ 𝑦 ∈ (𝐹‘𝑤)))
53		disj1 4405	. . . . . . . 8 ⊢ (((𝐹‘𝑧) ∩ (𝐹‘𝑤)) = ∅ ↔ ∀𝑦(𝑦 ∈ (𝐹‘𝑧) → ¬ 𝑦 ∈ (𝐹‘𝑤)))
54	52, 53	bitr4i 278	. . . . . . 7 ⊢ (∀𝑦(𝑦𝑅𝑧 → ¬ 𝑦𝑅𝑤) ↔ ((𝐹‘𝑧) ∩ (𝐹‘𝑤)) = ∅)
55	46, 54	imbi12i 350	. . . . . 6 ⊢ ((¬ 𝑧 = 𝑤 → ∀𝑦(𝑦𝑅𝑧 → ¬ 𝑦𝑅𝑤)) ↔ ((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ((𝐹‘𝑧) ∩ (𝐹‘𝑤)) = ∅))
56	38, 55	imbi12i 350	. . . . 5 ⊢ (((𝑧𝑅𝑥 ∧ 𝑤𝑅𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦𝑅𝑧 → ¬ 𝑦𝑅𝑤))) ↔ ((𝑧 ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ (𝐹‘𝑥)) → ((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ((𝐹‘𝑧) ∩ (𝐹‘𝑤)) = ∅)))
57	56	2albii 1822	. . . 4 ⊢ (∀𝑧∀𝑤((𝑧𝑅𝑥 ∧ 𝑤𝑅𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦𝑅𝑧 → ¬ 𝑦𝑅𝑤))) ↔ ∀𝑧∀𝑤((𝑧 ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ (𝐹‘𝑥)) → ((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ((𝐹‘𝑧) ∩ (𝐹‘𝑤)) = ∅)))
58	36, 57	bitr4i 278	. . 3 ⊢ (∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∀𝑞 ∈ (𝐹 “ (𝐹‘𝑥))(𝑡 ≠ 𝑞 → (𝑡 ∩ 𝑞) = ∅) ↔ ∀𝑧∀𝑤((𝑧𝑅𝑥 ∧ 𝑤𝑅𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦𝑅𝑧 → ¬ 𝑦𝑅𝑤))))
59		f1ofun 6777	. . . . 5 ⊢ (𝐹:V–1-1-onto→V → Fun 𝐹)
60		fvex 6848	. . . . . 6 ⊢ (𝐹‘𝑥) ∈ V
61	60	funimaex 6581	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 “ (𝐹‘𝑥)) ∈ V)
62	2, 59, 61	mp2b 10	. . . 4 ⊢ (𝐹 “ (𝐹‘𝑥)) ∈ V
63		raleq 3294	. . . . . 6 ⊢ (𝑟 = (𝐹 “ (𝐹‘𝑥)) → (∀𝑡 ∈ 𝑟 𝑡 ≠ ∅ ↔ ∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))𝑡 ≠ ∅))
64		raleq 3294	. . . . . . 7 ⊢ (𝑟 = (𝐹 “ (𝐹‘𝑥)) → (∀𝑞 ∈ 𝑟 (𝑡 ≠ 𝑞 → (𝑡 ∩ 𝑞) = ∅) ↔ ∀𝑞 ∈ (𝐹 “ (𝐹‘𝑥))(𝑡 ≠ 𝑞 → (𝑡 ∩ 𝑞) = ∅)))
65	64	raleqbi1dv 3309	. . . . . 6 ⊢ (𝑟 = (𝐹 “ (𝐹‘𝑥)) → (∀𝑡 ∈ 𝑟 ∀𝑞 ∈ 𝑟 (𝑡 ≠ 𝑞 → (𝑡 ∩ 𝑞) = ∅) ↔ ∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∀𝑞 ∈ (𝐹 “ (𝐹‘𝑥))(𝑡 ≠ 𝑞 → (𝑡 ∩ 𝑞) = ∅)))
66	63, 65	anbi12d 633	. . . . 5 ⊢ (𝑟 = (𝐹 “ (𝐹‘𝑥)) → ((∀𝑡 ∈ 𝑟 𝑡 ≠ ∅ ∧ ∀𝑡 ∈ 𝑟 ∀𝑞 ∈ 𝑟 (𝑡 ≠ 𝑞 → (𝑡 ∩ 𝑞) = ∅)) ↔ (∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))𝑡 ≠ ∅ ∧ ∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∀𝑞 ∈ (𝐹 “ (𝐹‘𝑥))(𝑡 ≠ 𝑞 → (𝑡 ∩ 𝑞) = ∅))))
67		raleq 3294	. . . . . 6 ⊢ (𝑟 = (𝐹 “ (𝐹‘𝑥)) → (∀𝑡 ∈ 𝑟 ∃!𝑣 𝑣 ∈ (𝑡 ∩ 𝑠) ↔ ∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∃!𝑣 𝑣 ∈ (𝑡 ∩ 𝑠)))
68	67	exbidv 1923	. . . . 5 ⊢ (𝑟 = (𝐹 “ (𝐹‘𝑥)) → (∃𝑠∀𝑡 ∈ 𝑟 ∃!𝑣 𝑣 ∈ (𝑡 ∩ 𝑠) ↔ ∃𝑠∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∃!𝑣 𝑣 ∈ (𝑡 ∩ 𝑠)))
69	66, 68	imbi12d 344	. . . 4 ⊢ (𝑟 = (𝐹 “ (𝐹‘𝑥)) → (((∀𝑡 ∈ 𝑟 𝑡 ≠ ∅ ∧ ∀𝑡 ∈ 𝑟 ∀𝑞 ∈ 𝑟 (𝑡 ≠ 𝑞 → (𝑡 ∩ 𝑞) = ∅)) → ∃𝑠∀𝑡 ∈ 𝑟 ∃!𝑣 𝑣 ∈ (𝑡 ∩ 𝑠)) ↔ ((∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))𝑡 ≠ ∅ ∧ ∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∀𝑞 ∈ (𝐹 “ (𝐹‘𝑥))(𝑡 ≠ 𝑞 → (𝑡 ∩ 𝑞) = ∅)) → ∃𝑠∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∃!𝑣 𝑣 ∈ (𝑡 ∩ 𝑠))))
70		ac8 10406	. . . 4 ⊢ ((∀𝑡 ∈ 𝑟 𝑡 ≠ ∅ ∧ ∀𝑡 ∈ 𝑟 ∀𝑞 ∈ 𝑟 (𝑡 ≠ 𝑞 → (𝑡 ∩ 𝑞) = ∅)) → ∃𝑠∀𝑡 ∈ 𝑟 ∃!𝑣 𝑣 ∈ (𝑡 ∩ 𝑠))
71	62, 69, 70	vtocl 3516	. . 3 ⊢ ((∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))𝑡 ≠ ∅ ∧ ∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∀𝑞 ∈ (𝐹 “ (𝐹‘𝑥))(𝑡 ≠ 𝑞 → (𝑡 ∩ 𝑞) = ∅)) → ∃𝑠∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∃!𝑣 𝑣 ∈ (𝑡 ∩ 𝑠))
72	20, 58, 71	syl2anbr 600	. 2 ⊢ ((∀𝑧(𝑧𝑅𝑥 → ∃𝑤 𝑤𝑅𝑧) ∧ ∀𝑧∀𝑤((𝑧𝑅𝑥 ∧ 𝑤𝑅𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦𝑅𝑧 → ¬ 𝑦𝑅𝑤)))) → ∃𝑠∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∃!𝑣 𝑣 ∈ (𝑡 ∩ 𝑠))
73		ineq1 4166	. . . . . . . . 9 ⊢ (𝑡 = (𝐹‘𝑧) → (𝑡 ∩ 𝑠) = ((𝐹‘𝑧) ∩ 𝑠))
74	73	eleq2d 2823	. . . . . . . 8 ⊢ (𝑡 = (𝐹‘𝑧) → (𝑣 ∈ (𝑡 ∩ 𝑠) ↔ 𝑣 ∈ ((𝐹‘𝑧) ∩ 𝑠)))
75	74	eubidv 2587	. . . . . . 7 ⊢ (𝑡 = (𝐹‘𝑧) → (∃!𝑣 𝑣 ∈ (𝑡 ∩ 𝑠) ↔ ∃!𝑣 𝑣 ∈ ((𝐹‘𝑧) ∩ 𝑠)))
76	75	ralima 7185	. . . . . 6 ⊢ ((𝐹 Fn V ∧ (𝐹‘𝑥) ⊆ V) → (∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∃!𝑣 𝑣 ∈ (𝑡 ∩ 𝑠) ↔ ∀𝑧 ∈ (𝐹‘𝑥)∃!𝑣 𝑣 ∈ ((𝐹‘𝑧) ∩ 𝑠)))
77	4, 5, 76	mp2an 693	. . . . 5 ⊢ (∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∃!𝑣 𝑣 ∈ (𝑡 ∩ 𝑠) ↔ ∀𝑧 ∈ (𝐹‘𝑥)∃!𝑣 𝑣 ∈ ((𝐹‘𝑧) ∩ 𝑠))
78		df-ral 3053	. . . . 5 ⊢ (∀𝑧 ∈ (𝐹‘𝑥)∃!𝑣 𝑣 ∈ ((𝐹‘𝑧) ∩ 𝑠) ↔ ∀𝑧(𝑧 ∈ (𝐹‘𝑥) → ∃!𝑣 𝑣 ∈ ((𝐹‘𝑧) ∩ 𝑠)))
79	77, 78	bitri 275	. . . 4 ⊢ (∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∃!𝑣 𝑣 ∈ (𝑡 ∩ 𝑠) ↔ ∀𝑧(𝑧 ∈ (𝐹‘𝑥) → ∃!𝑣 𝑣 ∈ ((𝐹‘𝑧) ∩ 𝑠)))
80		fvex 6848	. . . . 5 ⊢ (◡𝐹‘𝑠) ∈ V
81	12	a1i 11	. . . . . . 7 ⊢ (𝑦 = (◡𝐹‘𝑠) → (𝑧𝑅𝑥 ↔ 𝑧 ∈ (𝐹‘𝑥)))
82		vex 3445	. . . . . . . . . . . . . 14 ⊢ 𝑣 ∈ V
83	2, 9, 82, 10	brpermmodel 45280	. . . . . . . . . . . . 13 ⊢ (𝑣𝑅𝑧 ↔ 𝑣 ∈ (𝐹‘𝑧))
84	83	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑦 = (◡𝐹‘𝑠) → (𝑣𝑅𝑧 ↔ 𝑣 ∈ (𝐹‘𝑧)))
85		breq2 5103	. . . . . . . . . . . . 13 ⊢ (𝑦 = (◡𝐹‘𝑠) → (𝑣𝑅𝑦 ↔ 𝑣𝑅(◡𝐹‘𝑠)))
86		vex 3445	. . . . . . . . . . . . . 14 ⊢ 𝑠 ∈ V
87	2, 9, 82, 86	brpermmodelcnv 45281	. . . . . . . . . . . . 13 ⊢ (𝑣𝑅(◡𝐹‘𝑠) ↔ 𝑣 ∈ 𝑠)
88	85, 87	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝑦 = (◡𝐹‘𝑠) → (𝑣𝑅𝑦 ↔ 𝑣 ∈ 𝑠))
89	84, 88	anbi12d 633	. . . . . . . . . . 11 ⊢ (𝑦 = (◡𝐹‘𝑠) → ((𝑣𝑅𝑧 ∧ 𝑣𝑅𝑦) ↔ (𝑣 ∈ (𝐹‘𝑧) ∧ 𝑣 ∈ 𝑠)))
90	89	bibi1d 343	. . . . . . . . . 10 ⊢ (𝑦 = (◡𝐹‘𝑠) → (((𝑣𝑅𝑧 ∧ 𝑣𝑅𝑦) ↔ 𝑣 = 𝑤) ↔ ((𝑣 ∈ (𝐹‘𝑧) ∧ 𝑣 ∈ 𝑠) ↔ 𝑣 = 𝑤)))
91	90	albidv 1922	. . . . . . . . 9 ⊢ (𝑦 = (◡𝐹‘𝑠) → (∀𝑣((𝑣𝑅𝑧 ∧ 𝑣𝑅𝑦) ↔ 𝑣 = 𝑤) ↔ ∀𝑣((𝑣 ∈ (𝐹‘𝑧) ∧ 𝑣 ∈ 𝑠) ↔ 𝑣 = 𝑤)))
92	91	exbidv 1923	. . . . . . . 8 ⊢ (𝑦 = (◡𝐹‘𝑠) → (∃𝑤∀𝑣((𝑣𝑅𝑧 ∧ 𝑣𝑅𝑦) ↔ 𝑣 = 𝑤) ↔ ∃𝑤∀𝑣((𝑣 ∈ (𝐹‘𝑧) ∧ 𝑣 ∈ 𝑠) ↔ 𝑣 = 𝑤)))
93		elin 3918	. . . . . . . . . 10 ⊢ (𝑣 ∈ ((𝐹‘𝑧) ∩ 𝑠) ↔ (𝑣 ∈ (𝐹‘𝑧) ∧ 𝑣 ∈ 𝑠))
94	93	eubii 2586	. . . . . . . . 9 ⊢ (∃!𝑣 𝑣 ∈ ((𝐹‘𝑧) ∩ 𝑠) ↔ ∃!𝑣(𝑣 ∈ (𝐹‘𝑧) ∧ 𝑣 ∈ 𝑠))
95		eu6 2575	. . . . . . . . 9 ⊢ (∃!𝑣(𝑣 ∈ (𝐹‘𝑧) ∧ 𝑣 ∈ 𝑠) ↔ ∃𝑤∀𝑣((𝑣 ∈ (𝐹‘𝑧) ∧ 𝑣 ∈ 𝑠) ↔ 𝑣 = 𝑤))
96	94, 95	bitri 275	. . . . . . . 8 ⊢ (∃!𝑣 𝑣 ∈ ((𝐹‘𝑧) ∩ 𝑠) ↔ ∃𝑤∀𝑣((𝑣 ∈ (𝐹‘𝑧) ∧ 𝑣 ∈ 𝑠) ↔ 𝑣 = 𝑤))
97	92, 96	bitr4di 289	. . . . . . 7 ⊢ (𝑦 = (◡𝐹‘𝑠) → (∃𝑤∀𝑣((𝑣𝑅𝑧 ∧ 𝑣𝑅𝑦) ↔ 𝑣 = 𝑤) ↔ ∃!𝑣 𝑣 ∈ ((𝐹‘𝑧) ∩ 𝑠)))
98	81, 97	imbi12d 344	. . . . . 6 ⊢ (𝑦 = (◡𝐹‘𝑠) → ((𝑧𝑅𝑥 → ∃𝑤∀𝑣((𝑣𝑅𝑧 ∧ 𝑣𝑅𝑦) ↔ 𝑣 = 𝑤)) ↔ (𝑧 ∈ (𝐹‘𝑥) → ∃!𝑣 𝑣 ∈ ((𝐹‘𝑧) ∩ 𝑠))))
99	98	albidv 1922	. . . . 5 ⊢ (𝑦 = (◡𝐹‘𝑠) → (∀𝑧(𝑧𝑅𝑥 → ∃𝑤∀𝑣((𝑣𝑅𝑧 ∧ 𝑣𝑅𝑦) ↔ 𝑣 = 𝑤)) ↔ ∀𝑧(𝑧 ∈ (𝐹‘𝑥) → ∃!𝑣 𝑣 ∈ ((𝐹‘𝑧) ∩ 𝑠))))
100	80, 99	spcev 3561	. . . 4 ⊢ (∀𝑧(𝑧 ∈ (𝐹‘𝑥) → ∃!𝑣 𝑣 ∈ ((𝐹‘𝑧) ∩ 𝑠)) → ∃𝑦∀𝑧(𝑧𝑅𝑥 → ∃𝑤∀𝑣((𝑣𝑅𝑧 ∧ 𝑣𝑅𝑦) ↔ 𝑣 = 𝑤)))
101	79, 100	sylbi 217	. . 3 ⊢ (∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∃!𝑣 𝑣 ∈ (𝑡 ∩ 𝑠) → ∃𝑦∀𝑧(𝑧𝑅𝑥 → ∃𝑤∀𝑣((𝑣𝑅𝑧 ∧ 𝑣𝑅𝑦) ↔ 𝑣 = 𝑤)))
102	101	exlimiv 1932	. 2 ⊢ (∃𝑠∀𝑡 ∈ (𝐹 “ (𝐹‘𝑥))∃!𝑣 𝑣 ∈ (𝑡 ∩ 𝑠) → ∃𝑦∀𝑧(𝑧𝑅𝑥 → ∃𝑤∀𝑣((𝑣𝑅𝑧 ∧ 𝑣𝑅𝑦) ↔ 𝑣 = 𝑤)))
103	72, 102	syl 17	1 ⊢ ((∀𝑧(𝑧𝑅𝑥 → ∃𝑤 𝑤𝑅𝑧) ∧ ∀𝑧∀𝑤((𝑧𝑅𝑥 ∧ 𝑤𝑅𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦𝑅𝑧 → ¬ 𝑦𝑅𝑤)))) → ∃𝑦∀𝑧(𝑧𝑅𝑥 → ∃𝑤∀𝑣((𝑣𝑅𝑧 ∧ 𝑣𝑅𝑦) ↔ 𝑣 = 𝑤)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃!weu 2569   ≠ wne 2933  ∀wral 3052  Vcvv 3441   ∩ cin 3901   ⊆ wss 3902  ∅c0 4286   class class class wbr 5099   E cep 5524  ^◡ccnv 5624   “ cima 5628   ∘ ccom 5629  Fun wfun 6487   Fn wfn 6488  –1-1→wf1 6490  –1-1-onto→wf1o 6492  ‘cfv 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-ac2 10377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ac 10030

This theorem is referenced by: (None)