Theorem dfac11 40887

Description: The right-hand side of this theorem (compare with ac4 10231), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 9351, despite not mentioning the cumulative hierarchy in any way as most consequences of regularity do.

This is definition (MC) of [Schechter] p. 141. EDITORIAL: the proof is not original with me of course but I lost my reference sometime after writing it.

A multiple choice function allows any total order to be extended to a choice function, which in turn defines a well-ordering. Since a well-ordering on a set defines a simple ordering of the power set, this allows the trivial well-ordering of the empty set to be transfinitely bootstrapped up the cumulative hierarchy to any desired level. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Stefan O'Rear, 1-Jun-2015.)

Assertion

Ref	Expression
dfac11	⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))

Distinct variable group:   𝑥,𝑧,𝑓

Proof of Theorem dfac11

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfac3 9877	. . 3 ⊢ (CHOICE ↔ ∀𝑎∃𝑐∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑))
2		raleq 3342	. . . . . 6 ⊢ (𝑎 = 𝑥 → (∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ ∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑)))
3	2	exbidv 1924	. . . . 5 ⊢ (𝑎 = 𝑥 → (∃𝑐∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ ∃𝑐∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑)))
4	3	cbvalvw 2039	. . . 4 ⊢ (∀𝑎∃𝑐∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ ∀𝑥∃𝑐∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑))
5		neeq1 3006	. . . . . . . . . 10 ⊢ (𝑑 = 𝑧 → (𝑑 ≠ ∅ ↔ 𝑧 ≠ ∅))
6		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑧 → (𝑐‘𝑑) = (𝑐‘𝑧))
7		id 22	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑧 → 𝑑 = 𝑧)
8	6, 7	eleq12d 2833	. . . . . . . . . 10 ⊢ (𝑑 = 𝑧 → ((𝑐‘𝑑) ∈ 𝑑 ↔ (𝑐‘𝑧) ∈ 𝑧))
9	5, 8	imbi12d 345	. . . . . . . . 9 ⊢ (𝑑 = 𝑧 → ((𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ (𝑧 ≠ ∅ → (𝑐‘𝑧) ∈ 𝑧)))
10	9	cbvralvw 3383	. . . . . . . 8 ⊢ (∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑐‘𝑧) ∈ 𝑧))
11		fveq2 6774	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑧 → (𝑐‘𝑏) = (𝑐‘𝑧))
12	11	sneqd 4573	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑧 → {(𝑐‘𝑏)} = {(𝑐‘𝑧)})
13		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) = (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})
14		snex 5354	. . . . . . . . . . . . . 14 ⊢ {(𝑐‘𝑧)} ∈ V
15	12, 13, 14	fvmpt 6875	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑥 → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) = {(𝑐‘𝑧)})
16	15	3ad2ant1 1132	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) = {(𝑐‘𝑧)})
17		simp3 1137	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → (𝑐‘𝑧) ∈ 𝑧)
18	17	snssd 4742	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ⊆ 𝑧)
19	14	elpw 4537	. . . . . . . . . . . . . . 15 ⊢ ({(𝑐‘𝑧)} ∈ 𝒫 𝑧 ↔ {(𝑐‘𝑧)} ⊆ 𝑧)
20	18, 19	sylibr 233	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ∈ 𝒫 𝑧)
21		snfi 8834	. . . . . . . . . . . . . . 15 ⊢ {(𝑐‘𝑧)} ∈ Fin
22	21	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ∈ Fin)
23	20, 22	elind 4128	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ∈ (𝒫 𝑧 ∩ Fin))
24		fvex 6787	. . . . . . . . . . . . . . 15 ⊢ (𝑐‘𝑧) ∈ V
25	24	snnz 4712	. . . . . . . . . . . . . 14 ⊢ {(𝑐‘𝑧)} ≠ ∅
26	25	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ≠ ∅)
27		eldifsn 4720	. . . . . . . . . . . . 13 ⊢ ({(𝑐‘𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}) ↔ ({(𝑐‘𝑧)} ∈ (𝒫 𝑧 ∩ Fin) ∧ {(𝑐‘𝑧)} ≠ ∅))
28	23, 26, 27	sylanbrc 583	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))
29	16, 28	eqeltrd 2839	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))
30	29	3exp 1118	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ≠ ∅ → ((𝑐‘𝑧) ∈ 𝑧 → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
31	30	a2d 29	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑥 → ((𝑧 ≠ ∅ → (𝑐‘𝑧) ∈ 𝑧) → (𝑧 ≠ ∅ → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
32	31	ralimia 3085	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑐‘𝑧) ∈ 𝑧) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
33	10, 32	sylbi 216	. . . . . . 7 ⊢ (∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
34		vex 3436	. . . . . . . . 9 ⊢ 𝑥 ∈ V
35	34	mptex 7099	. . . . . . . 8 ⊢ (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) ∈ V
36		fveq1 6773	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) → (𝑓‘𝑧) = ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧))
37	36	eleq1d 2823	. . . . . . . . . 10 ⊢ (𝑓 = (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) → ((𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}) ↔ ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
38	37	imbi2d 341	. . . . . . . . 9 ⊢ (𝑓 = (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ (𝑧 ≠ ∅ → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
39	38	ralbidv 3112	. . . . . . . 8 ⊢ (𝑓 = (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
40	35, 39	spcev 3545	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
41	33, 40	syl 17	. . . . . 6 ⊢ (∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
42	41	exlimiv 1933	. . . . 5 ⊢ (∃𝑐∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
43	42	alimi 1814	. . . 4 ⊢ (∀𝑥∃𝑐∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
44	4, 43	sylbi 216	. . 3 ⊢ (∀𝑎∃𝑐∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
45	1, 44	sylbi 216	. 2 ⊢ (CHOICE → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
46		fvex 6787	. . . . . . 7 ⊢ (𝑅1‘(rank‘𝑎)) ∈ V
47	46	pwex 5303	. . . . . 6 ⊢ 𝒫 (𝑅1‘(rank‘𝑎)) ∈ V
48		raleq 3342	. . . . . . 7 ⊢ (𝑥 = 𝒫 (𝑅1‘(rank‘𝑎)) → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
49	48	exbidv 1924	. . . . . 6 ⊢ (𝑥 = 𝒫 (𝑅1‘(rank‘𝑎)) → (∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∃𝑓∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
50	47, 49	spcv 3544	. . . . 5 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑓∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
51		rankon 9553	. . . . . . . 8 ⊢ (rank‘𝑎) ∈ On
52	51	a1i 11	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → (rank‘𝑎) ∈ On)
53		id 22	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
54	52, 53	aomclem8 40886	. . . . . 6 ⊢ (∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
55	54	exlimiv 1933	. . . . 5 ⊢ (∃𝑓∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
56		vex 3436	. . . . . 6 ⊢ 𝑎 ∈ V
57		r1rankid 9617	. . . . . 6 ⊢ (𝑎 ∈ V → 𝑎 ⊆ (𝑅1‘(rank‘𝑎)))
58		wess 5576	. . . . . . 7 ⊢ (𝑎 ⊆ (𝑅1‘(rank‘𝑎)) → (𝑏 We (𝑅1‘(rank‘𝑎)) → 𝑏 We 𝑎))
59	58	eximdv 1920	. . . . . 6 ⊢ (𝑎 ⊆ (𝑅1‘(rank‘𝑎)) → (∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)) → ∃𝑏 𝑏 We 𝑎))
60	56, 57, 59	mp2b 10	. . . . 5 ⊢ (∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)) → ∃𝑏 𝑏 We 𝑎)
61	50, 55, 60	3syl 18	. . . 4 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We 𝑎)
62	61	alrimiv 1930	. . 3 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∀𝑎∃𝑏 𝑏 We 𝑎)
63		dfac8 9891	. . 3 ⊢ (CHOICE ↔ ∀𝑎∃𝑏 𝑏 We 𝑎)
64	62, 63	sylibr 233	. 2 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → CHOICE)
65	45, 64	impbii 208	1 ⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1086  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  Vcvv 3432   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561   ↦ cmpt 5157   We wwe 5543  Oncon0 6266  ‘cfv 6433  Fincfn 8733  𝑅₁cr1 9520  rankcrnk 9521  CHOICEwac 9871

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-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  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-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-map 8617  df-en 8734  df-fin 8737  df-sup 9201  df-r1 9522  df-rank 9523  df-card 9697  df-ac 9872

This theorem is referenced by: (None)