Theorem dfac11 43507

Description: The right-hand side of this theorem (compare with ac4 10388), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 9497, 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 10034	. . 3 ⊢ (CHOICE ↔ ∀𝑎∃𝑐∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑))
2		raleq 3294	. . . . . 6 ⊢ (𝑎 = 𝑥 → (∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ ∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑)))
3	2	exbidv 1928	. . . . 5 ⊢ (𝑎 = 𝑥 → (∃𝑐∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ ∃𝑐∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑)))
4	3	cbvalvw 2043	. . . 4 ⊢ (∀𝑎∃𝑐∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ ∀𝑥∃𝑐∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑))
5		neeq1 2996	. . . . . . . . . 10 ⊢ (𝑑 = 𝑧 → (𝑑 ≠ ∅ ↔ 𝑧 ≠ ∅))
6		fveq2 6827	. . . . . . . . . . 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 3217	. . . . . . . 8 ⊢ (∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) ↔ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑐‘𝑧) ∈ 𝑧))
11		fveq2 6827	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑧 → (𝑐‘𝑏) = (𝑐‘𝑧))
12	11	sneqd 4567	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑧 → {(𝑐‘𝑏)} = {(𝑐‘𝑧)})
13		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) = (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})
14		snex 5368	. . . . . . . . . . . . . 14 ⊢ {(𝑐‘𝑧)} ∈ V
15	12, 13, 14	fvmpt 6935	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑥 → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) = {(𝑐‘𝑧)})
16	15	3ad2ant1 1139	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) = {(𝑐‘𝑧)})
17		simp3 1144	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → (𝑐‘𝑧) ∈ 𝑧)
18	17	snssd 4718	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ⊆ 𝑧)
19	14	elpw 4533	. . . . . . . . . . . . . . 15 ⊢ ({(𝑐‘𝑧)} ∈ 𝒫 𝑧 ↔ {(𝑐‘𝑧)} ⊆ 𝑧)
20	18, 19	sylibr 235	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ∈ 𝒫 𝑧)
21		snfi 8980	. . . . . . . . . . . . . . 15 ⊢ {(𝑐‘𝑧)} ∈ Fin
22	21	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ∈ Fin)
23	20, 22	elind 4129	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ∈ (𝒫 𝑧 ∩ Fin))
24		fvex 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑐‘𝑧) ∈ V
25	24	snnz 4708	. . . . . . . . . . . . . 14 ⊢ {(𝑐‘𝑧)} ≠ ∅
26	25	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ∧ (𝑐‘𝑧) ∈ 𝑧) → {(𝑐‘𝑧)} ≠ ∅)
27		eldifsn 4719	. . . . . . . . . . . . 13 ⊢ ({(𝑐‘𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}) ↔ ({(𝑐‘𝑧)} ∈ (𝒫 𝑧 ∩ Fin) ∧ {(𝑐‘𝑧)} ≠ ∅))
28	23, 26, 27	sylanbrc 589	. . . . . . . . . . . 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 1125	. . . . . . . . . 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 3073	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑐‘𝑧) ∈ 𝑧) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
33	10, 32	sylbi 218	. . . . . . 7 ⊢ (∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
34		vex 3435	. . . . . . . . 9 ⊢ 𝑥 ∈ V
35	34	mptex 7167	. . . . . . . 8 ⊢ (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) ∈ V
36		fveq1 6826	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)}) → (𝑓‘𝑧) = ((𝑏 ∈ 𝑥 ↦ {(𝑐‘𝑏)})‘𝑧))
37	36	eleq1d 2824	. . . . . . . . . 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 3162	. . . . . . . 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 3544	. . . . . . 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 1937	. . . . 5 ⊢ (∃𝑐∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
43	42	alimi 1818	. . . 4 ⊢ (∀𝑥∃𝑐∀𝑑 ∈ 𝑥 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
44	4, 43	sylbi 218	. . 3 ⊢ (∀𝑎∃𝑐∀𝑑 ∈ 𝑎 (𝑑 ≠ ∅ → (𝑐‘𝑑) ∈ 𝑑) → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
45	1, 44	sylbi 218	. 2 ⊢ (CHOICE → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
46		fvex 6840	. . . . . . 7 ⊢ (𝑅1‘(rank‘𝑎)) ∈ V
47	46	pwex 5309	. . . . . 6 ⊢ 𝒫 (𝑅1‘(rank‘𝑎)) ∈ V
48		raleq 3294	. . . . . . 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 1928	. . . . . 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 3543	. . . . 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 9710	. . . . . . . 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 43506	. . . . . 6 ⊢ (∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
55	54	exlimiv 1937	. . . . 5 ⊢ (∃𝑓∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
56		vex 3435	. . . . . 6 ⊢ 𝑎 ∈ V
57		r1rankid 9774	. . . . . 6 ⊢ (𝑎 ∈ V → 𝑎 ⊆ (𝑅1‘(rank‘𝑎)))
58		wess 5604	. . . . . . 7 ⊢ (𝑎 ⊆ (𝑅1‘(rank‘𝑎)) → (𝑏 We (𝑅1‘(rank‘𝑎)) → 𝑏 We 𝑎))
59	58	eximdv 1924	. . . . . 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 1934	. . 3 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∀𝑎∃𝑏 𝑏 We 𝑎)
63		dfac8 10049	. . 3 ⊢ (CHOICE ↔ ∀𝑎∃𝑏 𝑏 We 𝑎)
64	62, 63	sylibr 235	. 2 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → CHOICE)
65	45, 64	impbii 210	1 ⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))

Syntax hints:   → wi 4   ↔ wb 207   ∧ w3a 1092  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  Vcvv 3431   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  𝒫 cpw 4529  {csn 4555   ↦ cmpt 5153   We wwe 5570  Oncon0 6310  ‘cfv 6485  Fincfn 8883  𝑅₁cr1 9677  rankcrnk 9678  CHOICEwac 10028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-reg 9497  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-map 8765  df-en 8884  df-fin 8887  df-sup 9345  df-r1 9679  df-rank 9680  df-card 9854  df-ac 10029

This theorem is referenced by: (None)