Theorem dfac11 44204

Description: The right-hand side of this theorem (compare with ac4 10563), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 9596, 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 10210	. . 3 ⊢ (CHOICE ↔ ∀𝑎∃𝑐∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑))
2		raleq 3319	. . . . . 6 ⊢ (𝑎 = 𝑥 → (∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ ∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑)))
3	2	exbidv 1955	. . . . 5 ⊢ (𝑎 = 𝑥 → (∃𝑐∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ ∃𝑐∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑)))
4	3	cbvalvw 2070	. . . 4 ⊢ (∀𝑎∃𝑐∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ ∀𝑥∃𝑐∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑))
5		neeq1 3020	. . . . . . . . . 10 ⊢ (𝑑 = 𝑧 → (𝑑 ≠ ∅ ↔ 𝑧 ≠ ∅))
6		fveq2 6893	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑧 → (𝑐‘𝑑) = (𝑐‘𝑧))
7		id 23	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑧 → 𝑑 = 𝑧)
8	6, 7	eleq12d 2857	. . . . . . . . . 10 ⊢ (𝑑 = 𝑧 → ((𝑐‘𝑑) ∈ 𝑑 ↔ (𝑐‘𝑧) ∈ 𝑧))
9	5, 8	imbi12d 347	. . . . . . . . 9 ⊢ (𝑑 = 𝑧 → ((𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ (𝑧 ≠ ∅ → (𝑐‘𝑧) ∈ 𝑧)))
10	9	cbvralvw 3243	. . . . . . . 8 ⊢ (∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑐‘𝑧) ∈ 𝑧))
11		fveq2 6893	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑧 → (𝑐‘𝑏) = (𝑐‘𝑧))
12	11	sneqd 4595	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑧 → {(𝑐‘𝑏)} = {(𝑐‘𝑧)})
13		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) = (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})
14		snex 5400	. . . . . . . . . . . . . 14 ⊢ {(𝑐‘𝑧)} ∈ V
15	12, 13, 14	fvmpt 7001	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑥 → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) = {(𝑐‘𝑧)})
16	15	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) = {(𝑐‘𝑧)})
17		simp3 1156	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → (𝑐‘𝑧) ∈ 𝑧)
18	17	snssd 4746	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ⊆ 𝑧)
19	14	elpw 4563	. . . . . . . . . . . . . . 15 ⊢ ({(𝑐‘𝑧)} ∈ 𝒫 𝑧 ↔ {(𝑐‘𝑧)} ⊆ 𝑧)
20	18, 19	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ∈ 𝒫 𝑧)
21		snfi 9082	. . . . . . . . . . . . . . 15 ⊢ {(𝑐‘𝑧)} ∈ Fin
22	21	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ∈ Fin)
23	20, 22	elind 4099	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ∈ (𝒫 𝑧 ∩ Fin))
24		fvex 6906	. . . . . . . . . . . . . . 15 ⊢ (𝑐‘𝑧) ∈ V
25	24	snnz 4736	. . . . . . . . . . . . . 14 ⊢ {(𝑐‘𝑧)} ≠ ∅
26	25	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ≠ ∅)
27		eldifsn 4747	. . . . . . . . . . . . 13 ⊢ ({(𝑐‘𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}) ↔ ({(𝑐‘𝑧)} ∈ (𝒫 𝑧 ∩ Fin) ∧ {(𝑐‘𝑧)} ≠ ∅))
28	23, 26, 27	sylanbrc 595	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))
29	16, 28	eqeltrd 2863	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))
30	29	3exp 1137	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ≠ ∅ → ((𝑐‘𝑧) ∈ 𝑧 → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
31	30	a2d 30	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑥 → ((𝑧 ≠ ∅ → (𝑐‘𝑧) ∈ 𝑧) → (𝑧 ≠ ∅ → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
32	31	ralimia 3099	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑐‘𝑧) ∈ 𝑧) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
33	10, 32	sylbi 220	. . . . . . 7 ⊢ (∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
34		vex 3457	. . . . . . . . 9 ⊢ 𝑥 ∈ V
35	34	mptex 7236	. . . . . . . 8 ⊢ (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) ∈ V
36		fveq1 6892	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) → (𝑓‘𝑧) = ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧))
37	36	eleq1d 2848	. . . . . . . . . 10 ⊢ (𝑓 = (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) → ((𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}) ↔ ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
38	37	imbi2d 343	. . . . . . . . 9 ⊢ (𝑓 = (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ (𝑧 ≠ ∅ → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
39	38	ralbidv 3188	. . . . . . . 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 3563	. . . . . . 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 18	. . . . . 6 ⊢ (∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
42	41	exlimiv 1964	. . . . 5 ⊢ (∃𝑐∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
43	42	alimi 1843	. . . 4 ⊢ (∀𝑥∃𝑐∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
44	4, 43	sylbi 220	. . 3 ⊢ (∀𝑎∃𝑐∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
45	1, 44	sylbi 220	. 2 ⊢ (CHOICE → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
46		fvex 6906	. . . . . . 7 ⊢ (𝑅1‘(rank‘𝑎)) ∈ V
47	46	pwex 5345	. . . . . 6 ⊢ 𝒫 (𝑅1‘(rank‘𝑎)) ∈ V
48		raleq 3319	. . . . . . 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 1955	. . . . . 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 3562	. . . . 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 9813	. . . . . . . 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 23	. . . . . . 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 44203	. . . . . 6 ⊢ (∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
55	54	exlimiv 1964	. . . . 5 ⊢ (∃𝑓∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
56		vex 3457	. . . . . 6 ⊢ 𝑎 ∈ V
57		r1rankid 9885	. . . . . 6 ⊢ (𝑎 ∈ V → 𝑎 ⊆ (𝑅1‘(rank‘𝑎)))
58		wess 5640	. . . . . . 7 ⊢ (𝑎 ⊆ (𝑅1‘(rank‘𝑎)) → (𝑏 We (𝑅1‘(rank‘𝑎)) → 𝑏 We 𝑎))
59	58	eximdv 1951	. . . . . 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 19	. . . 4 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We 𝑎)
62	61	alrimiv 1961	. . 3 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∀𝑎∃𝑏 𝑏 We 𝑎)
63		dfac8 10224	. . 3 ⊢ (CHOICE ↔ ∀𝑎∃𝑏 𝑏 We 𝑎)
64	62, 63	sylibr 237	. 2 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → CHOICE)
65	45, 64	impbii 212	1 ⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2146   ≠ wne 2958  ∀wral 3079  Vcvv 3453   ⊆ wss 3899   ∖ cdif 4047   ∩ cin 4091  ∅c0 4280  𝒫 cpw 4559  {csn 4587   ↦ cmpt 5188   We wwe 5606  Oncon0 6368  ‘cfv 6546  Fincfn 8984  𝑅₁cr1 9776  rankcrnk 9777  CHOICEwac 10204

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-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-reg 9596  ax-inf2 9652

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-se 5608  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-isom 6555  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-1st 8008  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-2o 8488  df-map 8860  df-en 8985  df-fin 8988  df-sup 9444  df-r1 9778  df-rank 9779  df-card 10030  df-ac 10205

This theorem is used by: (None)