Theorem dfac5 9548

Description: Equivalence of two versions of the Axiom of Choice. The right-hand side is Theorem 6M(4) of [Enderton] p. 151 and asserts that given a family of mutually disjoint nonempty sets, a set exists containing exactly one member from each set in the family. The proof does not depend on AC. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.)

Assertion

Ref	Expression
dfac5	⊢ (CHOICE ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))

Distinct variable group:   𝑥,𝑧,𝑦,𝑤,𝑣

Proof of Theorem dfac5

Dummy variables 𝑓 ℎ 𝑢 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfac4 9542	. . 3 ⊢ (CHOICE ↔ ∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)))
2		neeq1 3078	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (𝑧 ≠ ∅ ↔ 𝑤 ≠ ∅))
3	2	cbvralvw 3450	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ↔ ∀𝑤 ∈ 𝑥 𝑤 ≠ ∅)
4	3	anbi2i 624	. . . . . . . . . . 11 ⊢ ((∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑧 ∈ 𝑥 𝑧 ≠ ∅) ↔ (∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑤 ∈ 𝑥 𝑤 ≠ ∅))
5		r19.26 3170	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ 𝑥 ((𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) ↔ (∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑤 ∈ 𝑥 𝑤 ≠ ∅))
6	4, 5	bitr4i 280	. . . . . . . . . 10 ⊢ ((∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑧 ∈ 𝑥 𝑧 ≠ ∅) ↔ ∀𝑤 ∈ 𝑥 ((𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅))
7		pm3.35 801	. . . . . . . . . . . 12 ⊢ ((𝑤 ≠ ∅ ∧ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)) → (𝑓‘𝑤) ∈ 𝑤)
8	7	ancoms 461	. . . . . . . . . . 11 ⊢ (((𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) → (𝑓‘𝑤) ∈ 𝑤)
9	8	ralimi 3160	. . . . . . . . . 10 ⊢ (∀𝑤 ∈ 𝑥 ((𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) → ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤)
10	6, 9	sylbi 219	. . . . . . . . 9 ⊢ ((∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑧 ∈ 𝑥 𝑧 ≠ ∅) → ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤)
11		r19.26 3170	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) ↔ (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)))
12		elin 4169	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ ran 𝑓))
13		fvelrnb 6721	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 Fn 𝑥 → (𝑣 ∈ ran 𝑓 ↔ ∃𝑡 ∈ 𝑥 (𝑓‘𝑡) = 𝑣))
14	13	biimpac 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑣 ∈ ran 𝑓 ∧ 𝑓 Fn 𝑥) → ∃𝑡 ∈ 𝑥 (𝑓‘𝑡) = 𝑣)
15		fveq2 6665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = 𝑡 → (𝑓‘𝑤) = (𝑓‘𝑡))
16		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = 𝑡 → 𝑤 = 𝑡)
17	15, 16	eleq12d 2907	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 = 𝑡 → ((𝑓‘𝑤) ∈ 𝑤 ↔ (𝑓‘𝑡) ∈ 𝑡))
18		neeq2 3079	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = 𝑡 → (𝑧 ≠ 𝑤 ↔ 𝑧 ≠ 𝑡))
19		ineq2 4183	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = 𝑡 → (𝑧 ∩ 𝑤) = (𝑧 ∩ 𝑡))
20	19	eqeq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = 𝑡 → ((𝑧 ∩ 𝑤) = ∅ ↔ (𝑧 ∩ 𝑡) = ∅))
21	18, 20	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 = 𝑡 → ((𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)))
22	17, 21	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑤 = 𝑡 → (((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) ↔ ((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅))))
23	22	rspcv 3618	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 ∈ 𝑥 → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅))))
24		minel 4415	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ∩ 𝑡) = ∅) → ¬ (𝑓‘𝑡) ∈ 𝑧)
25	24	ex 415	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓‘𝑡) ∈ 𝑡 → ((𝑧 ∩ 𝑡) = ∅ → ¬ (𝑓‘𝑡) ∈ 𝑧))
26	25	imim2d 57	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓‘𝑡) ∈ 𝑡 → ((𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅) → (𝑧 ≠ 𝑡 → ¬ (𝑓‘𝑡) ∈ 𝑧)))
27	26	imp 409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) → (𝑧 ≠ 𝑡 → ¬ (𝑓‘𝑡) ∈ 𝑧))
28	27	necon4ad 3035	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) → ((𝑓‘𝑡) ∈ 𝑧 → 𝑧 = 𝑡))
29		eleq1 2900	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓‘𝑡) = 𝑣 → ((𝑓‘𝑡) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧))
30	29	biimpar 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓‘𝑡) = 𝑣 ∧ 𝑣 ∈ 𝑧) → (𝑓‘𝑡) ∈ 𝑧)
31	28, 30	impel 508	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) ∧ ((𝑓‘𝑡) = 𝑣 ∧ 𝑣 ∈ 𝑧)) → 𝑧 = 𝑡)
32		fveq2 6665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = 𝑡 → (𝑓‘𝑧) = (𝑓‘𝑡))
33		eqeq2 2833	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓‘𝑡) = 𝑣 → ((𝑓‘𝑧) = (𝑓‘𝑡) ↔ (𝑓‘𝑧) = 𝑣))
34		eqcom 2828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓‘𝑧) = 𝑣 ↔ 𝑣 = (𝑓‘𝑧))
35	33, 34	syl6bb 289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓‘𝑡) = 𝑣 → ((𝑓‘𝑧) = (𝑓‘𝑡) ↔ 𝑣 = (𝑓‘𝑧)))
36	32, 35	syl5ib 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓‘𝑡) = 𝑣 → (𝑧 = 𝑡 → 𝑣 = (𝑓‘𝑧)))
37	36	ad2antrl 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) ∧ ((𝑓‘𝑡) = 𝑣 ∧ 𝑣 ∈ 𝑧)) → (𝑧 = 𝑡 → 𝑣 = (𝑓‘𝑧)))
38	31, 37	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) ∧ ((𝑓‘𝑡) = 𝑣 ∧ 𝑣 ∈ 𝑧)) → 𝑣 = (𝑓‘𝑧))
39	38	exp32 423	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) → ((𝑓‘𝑡) = 𝑣 → (𝑣 ∈ 𝑧 → 𝑣 = (𝑓‘𝑧))))
40	23, 39	syl6com 37	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑡 ∈ 𝑥 → ((𝑓‘𝑡) = 𝑣 → (𝑣 ∈ 𝑧 → 𝑣 = (𝑓‘𝑧)))))
41	40	com14 96	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 ∈ 𝑧 → (𝑡 ∈ 𝑥 → ((𝑓‘𝑡) = 𝑣 → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → 𝑣 = (𝑓‘𝑧)))))
42	41	rexlimdv 3283	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 ∈ 𝑧 → (∃𝑡 ∈ 𝑥 (𝑓‘𝑡) = 𝑣 → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → 𝑣 = (𝑓‘𝑧))))
43	14, 42	syl5 34	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ 𝑧 → ((𝑣 ∈ ran 𝑓 ∧ 𝑓 Fn 𝑥) → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → 𝑣 = (𝑓‘𝑧))))
44	43	expd 418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ 𝑧 → (𝑣 ∈ ran 𝑓 → (𝑓 Fn 𝑥 → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → 𝑣 = (𝑓‘𝑧)))))
45	44	com4t 93	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 Fn 𝑥 → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑣 ∈ 𝑧 → (𝑣 ∈ ran 𝑓 → 𝑣 = (𝑓‘𝑧)))))
46	45	imp4b 424	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ ran 𝑓) → 𝑣 = (𝑓‘𝑧)))
47	12, 46	syl5bi 244	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓‘𝑧)))
48	11, 47	sylan2br 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 Fn 𝑥 ∧ (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓‘𝑧)))
49	48	anassrs 470	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓‘𝑧)))
50	49	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓‘𝑧)))
51		fveq2 6665	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑧 → (𝑓‘𝑤) = (𝑓‘𝑧))
52		id 22	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑧 → 𝑤 = 𝑧)
53	51, 52	eleq12d 2907	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑧 → ((𝑓‘𝑤) ∈ 𝑤 ↔ (𝑓‘𝑧) ∈ 𝑧))
54	53	rspcv 3618	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ 𝑥 → (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 → (𝑓‘𝑧) ∈ 𝑧))
55		fnfvelrn 6843	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓 Fn 𝑥 ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ ran 𝑓)
56	55	expcom 416	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ 𝑥 → (𝑓 Fn 𝑥 → (𝑓‘𝑧) ∈ ran 𝑓))
57	54, 56	anim12d 610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝑥 → ((∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 ∧ 𝑓 Fn 𝑥) → ((𝑓‘𝑧) ∈ 𝑧 ∧ (𝑓‘𝑧) ∈ ran 𝑓)))
58		elin 4169	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓) ↔ ((𝑓‘𝑧) ∈ 𝑧 ∧ (𝑓‘𝑧) ∈ ran 𝑓))
59	57, 58	syl6ibr 254	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝑥 → ((∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 ∧ 𝑓 Fn 𝑥) → (𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓)))
60	59	expd 418	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑥 → (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 → (𝑓 Fn 𝑥 → (𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓))))
61	60	com13 88	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 Fn 𝑥 → (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 → (𝑧 ∈ 𝑥 → (𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓))))
62	61	imp31 420	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓))
63		eleq1 2900	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑓‘𝑧) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ (𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓)))
64	62, 63	syl5ibrcom 249	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) → (𝑣 = (𝑓‘𝑧) → 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
65	64	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑣 = (𝑓‘𝑧) → 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
66	50, 65	impbid 214	. . . . . . . . . . . . . 14 ⊢ ((((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧)))
67	66	ex 415	. . . . . . . . . . . . 13 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) → (∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧))))
68	67	alrimdv 1926	. . . . . . . . . . . 12 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) → (∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧))))
69		fvex 6678	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑧) ∈ V
70		eqeq2 2833	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑓‘𝑧) → (𝑣 = ℎ ↔ 𝑣 = (𝑓‘𝑧)))
71	70	bibi2d 345	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑓‘𝑧) → ((𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ℎ) ↔ (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧))))
72	71	albidv 1917	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝑓‘𝑧) → (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ℎ) ↔ ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧))))
73	69, 72	spcev 3607	. . . . . . . . . . . . 13 ⊢ (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧)) → ∃ℎ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ℎ))
74		eu6 2655	. . . . . . . . . . . . 13 ⊢ (∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ ∃ℎ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ℎ))
75	73, 74	sylibr 236	. . . . . . . . . . . 12 ⊢ (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧)) → ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))
76	68, 75	syl6 35	. . . . . . . . . . 11 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) → (∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
77	76	ralimdva 3177	. . . . . . . . . 10 ⊢ ((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) → (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
78	77	ex 415	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑥 → (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 → (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))))
79	10, 78	syl5 34	. . . . . . . 8 ⊢ (𝑓 Fn 𝑥 → ((∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑧 ∈ 𝑥 𝑧 ≠ ∅) → (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))))
80	79	expd 418	. . . . . . 7 ⊢ (𝑓 Fn 𝑥 → (∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) → (∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ → (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))))
81	80	imp4b 424	. . . . . 6 ⊢ ((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)) → ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
82		vex 3498	. . . . . . . 8 ⊢ 𝑓 ∈ V
83	82	rnex 7611	. . . . . . 7 ⊢ ran 𝑓 ∈ V
84		ineq2 4183	. . . . . . . . . 10 ⊢ (𝑦 = ran 𝑓 → (𝑧 ∩ 𝑦) = (𝑧 ∩ ran 𝑓))
85	84	eleq2d 2898	. . . . . . . . 9 ⊢ (𝑦 = ran 𝑓 → (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
86	85	eubidv 2668	. . . . . . . 8 ⊢ (𝑦 = ran 𝑓 → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
87	86	ralbidv 3197	. . . . . . 7 ⊢ (𝑦 = ran 𝑓 → (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
88	83, 87	spcev 3607	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
89	81, 88	syl6 35	. . . . 5 ⊢ ((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)) → ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
90	89	exlimiv 1927	. . . 4 ⊢ (∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)) → ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
91	90	alimi 1808	. . 3 ⊢ (∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)) → ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
92	1, 91	sylbi 219	. 2 ⊢ (CHOICE → ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
93		eqid 2821	. . . . 5 ⊢ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))} = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))}
94		eqid 2821	. . . . 5 ⊢ (∪ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))} ∩ 𝑦) = (∪ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))} ∩ 𝑦)
95		biid 263	. . . . 5 ⊢ (∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
96	93, 94, 95	dfac5lem5 9547	. . . 4 ⊢ (∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) → ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
97	96	alrimiv 1924	. . 3 ⊢ (∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) → ∀ℎ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
98		dfac3 9541	. . 3 ⊢ (CHOICE ↔ ∀ℎ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
99	97, 98	sylibr 236	. 2 ⊢ (∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) → CHOICE)
100	92, 99	impbii 211	1 ⊢ (CHOICE ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   = wceq 1533  ∃wex 1776   ∈ wcel 2110  ∃!weu 2649  {cab 2799   ≠ wne 3016  ∀wral 3138  ∃wrex 3139   ∩ cin 3935  ∅c0 4291  {csn 4561  ∪ cuni 4832   × cxp 5548  ran crn 5551   Fn wfn 6345  ‘cfv 6350  CHOICEwac 9535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-ac 9536

This theorem is referenced by:  dfackm  9586  ac8  9908