Theorem dfac11 36650

Description: The right-hand side of this theorem (compare with ac4 9180), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 8380, 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 8827	. . 3 ⊢ (CHOICE ↔ ∀𝑎∃𝑐∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑))
2		raleq 3115	. . . . . 6 ⊢ (𝑎 = 𝑥 → (∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ ∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑)))
3	2	exbidv 1837	. . . . 5 ⊢ (𝑎 = 𝑥 → (∃𝑐∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ ∃𝑐∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑)))
4	3	cbvalv 2261	. . . 4 ⊢ (∀𝑎∃𝑐∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ ∀𝑥∃𝑐∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑))
5		neeq1 2844	. . . . . . . . . 10 ⊢ (𝑑 = 𝑧 → (𝑑 ≠ ∅ ↔ 𝑧 ≠ ∅))
6		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑧 → (𝑐‘𝑑) = (𝑐‘𝑧))
7		id 22	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑧 → 𝑑 = 𝑧)
8	6, 7	eleq12d 2682	. . . . . . . . . 10 ⊢ (𝑑 = 𝑧 → ((𝑐‘𝑑) ∈ 𝑑 ↔ (𝑐‘𝑧) ∈ 𝑧))
9	5, 8	imbi12d 333	. . . . . . . . 9 ⊢ (𝑑 = 𝑧 → ((𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ (𝑧 ≠ ∅ → (𝑐‘𝑧) ∈ 𝑧)))
10	9	cbvralv 3147	. . . . . . . 8 ⊢ (∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑐‘𝑧) ∈ 𝑧))
11		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑧 → (𝑐‘𝑏) = (𝑐‘𝑧))
12	11	sneqd 4137	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑧 → {(𝑐‘𝑏)} = {(𝑐‘𝑧)})
13		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) = (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})
14		snex 4835	. . . . . . . . . . . . . 14 ⊢ {(𝑐‘𝑧)} ∈ V
15	12, 13, 14	fvmpt 6191	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑥 → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) = {(𝑐‘𝑧)})
16	15	3ad2ant1 1075	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) = {(𝑐‘𝑧)})
17		simp3 1056	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → (𝑐‘𝑧) ∈ 𝑧)
18	17	snssd 4281	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ⊆ 𝑧)
19	14	elpw 4114	. . . . . . . . . . . . . . 15 ⊢ ({(𝑐‘𝑧)} ∈ 𝒫 𝑧 ↔ {(𝑐‘𝑧)} ⊆ 𝑧)
20	18, 19	sylibr 223	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ∈ 𝒫 𝑧)
21		snfi 7923	. . . . . . . . . . . . . . 15 ⊢ {(𝑐‘𝑧)} ∈ Fin
22	21	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ∈ Fin)
23	20, 22	elind 3760	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ∈ (𝒫 𝑧 ∩ Fin))
24		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ (𝑐‘𝑧) ∈ V
25	24	snnz 4252	. . . . . . . . . . . . . 14 ⊢ {(𝑐‘𝑧)} ≠ ∅
26	25	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ≠ ∅)
27		eldifsn 4260	. . . . . . . . . . . . 13 ⊢ ({(𝑐‘𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}) ↔ ({(𝑐‘𝑧)} ∈ (𝒫 𝑧 ∩ Fin) ∧ {(𝑐‘𝑧)} ≠ ∅))
28	23, 26, 27	sylanbrc 695	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))
29	16, 28	eqeltrd 2688	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))
30	29	3exp 1256	. . . . . . . . . 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 2934	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑐‘𝑧) ∈ 𝑧) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
33	10, 32	sylbi 206	. . . . . . 7 ⊢ (∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
34		vex 3176	. . . . . . . . 9 ⊢ 𝑥 ∈ V
35	34	mptex 6390	. . . . . . . 8 ⊢ (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) ∈ V
36		fveq1 6102	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) → (𝑓‘𝑧) = ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧))
37	36	eleq1d 2672	. . . . . . . . . 10 ⊢ (𝑓 = (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) → ((𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}) ↔ ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
38	37	imbi2d 329	. . . . . . . . 9 ⊢ (𝑓 = (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ (𝑧 ≠ ∅ → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
39	38	ralbidv 2969	. . . . . . . 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 3273	. . . . . . 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 1845	. . . . 5 ⊢ (∃𝑐∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
43	42	alimi 1730	. . . 4 ⊢ (∀𝑥∃𝑐∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
44	4, 43	sylbi 206	. . 3 ⊢ (∀𝑎∃𝑐∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
45	1, 44	sylbi 206	. 2 ⊢ (CHOICE → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
46		fvex 6113	. . . . . . 7 ⊢ (𝑅1‘(rank‘𝑎)) ∈ V
47	46	pwex 4774	. . . . . 6 ⊢ 𝒫 (𝑅1‘(rank‘𝑎)) ∈ V
48		raleq 3115	. . . . . . 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 1837	. . . . . 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 3272	. . . . 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 8541	. . . . . . . 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 36649	. . . . . 6 ⊢ (∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
55	54	exlimiv 1845	. . . . 5 ⊢ (∃𝑓∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
56		vex 3176	. . . . . 6 ⊢ 𝑎 ∈ V
57		r1rankid 8605	. . . . . 6 ⊢ (𝑎 ∈ V → 𝑎 ⊆ (𝑅1‘(rank‘𝑎)))
58		wess 5025	. . . . . . 7 ⊢ (𝑎 ⊆ (𝑅1‘(rank‘𝑎)) → (𝑏 We (𝑅1‘(rank‘𝑎)) → 𝑏 We 𝑎))
59	58	eximdv 1833	. . . . . 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 1842	. . 3 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∀𝑎∃𝑏 𝑏 We 𝑎)
63		dfac8 8840	. . 3 ⊢ (CHOICE ↔ ∀𝑎∃𝑏 𝑏 We 𝑎)
64	62, 63	sylibr 223	. 2 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → CHOICE)
65	45, 64	impbii 198	1 ⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))

Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125   ↦ cmpt 4643   We wwe 4996  Oncon0 5640  ‘cfv 5804  Fincfn 7841  𝑅₁cr1 8508  rankcrnk 8509  CHOICEwac 8821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-er 7629  df-map 7746  df-en 7842  df-fin 7845  df-sup 8231  df-r1 8510  df-rank 8511  df-card 8648  df-ac 8822

This theorem is referenced by: (None)