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Theorem dfac5 8989
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((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
Distinct variable group:   𝑥,𝑧,𝑦,𝑤,𝑣

Proof of Theorem dfac5
Dummy variables 𝑓 𝑢 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfac4 8983 . . 3 (CHOICE ↔ ∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)))
2 neeq1 2885 . . . . . . . . . . . . 13 (𝑧 = 𝑤 → (𝑧 ≠ ∅ ↔ 𝑤 ≠ ∅))
32cbvralv 3201 . . . . . . . . . . . 12 (∀𝑧𝑥 𝑧 ≠ ∅ ↔ ∀𝑤𝑥 𝑤 ≠ ∅)
43anbi2i 730 . . . . . . . . . . 11 ((∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑧𝑥 𝑧 ≠ ∅) ↔ (∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑤𝑥 𝑤 ≠ ∅))
5 r19.26 3093 . . . . . . . . . . 11 (∀𝑤𝑥 ((𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) ↔ (∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑤𝑥 𝑤 ≠ ∅))
64, 5bitr4i 267 . . . . . . . . . 10 ((∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑧𝑥 𝑧 ≠ ∅) ↔ ∀𝑤𝑥 ((𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅))
7 pm3.35 610 . . . . . . . . . . . 12 ((𝑤 ≠ ∅ ∧ (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)) → (𝑓𝑤) ∈ 𝑤)
87ancoms 468 . . . . . . . . . . 11 (((𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) → (𝑓𝑤) ∈ 𝑤)
98ralimi 2981 . . . . . . . . . 10 (∀𝑤𝑥 ((𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) → ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤)
106, 9sylbi 207 . . . . . . . . 9 ((∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑧𝑥 𝑧 ≠ ∅) → ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤)
11 r19.26 3093 . . . . . . . . . . . . . . . . . 18 (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) ↔ (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)))
12 elin 3829 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ (𝑣𝑧𝑣 ∈ ran 𝑓))
13 fvelrnb 6282 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 Fn 𝑥 → (𝑣 ∈ ran 𝑓 ↔ ∃𝑡𝑥 (𝑓𝑡) = 𝑣))
1413biimpac 502 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑣 ∈ ran 𝑓𝑓 Fn 𝑥) → ∃𝑡𝑥 (𝑓𝑡) = 𝑣)
15 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = 𝑡 → (𝑓𝑤) = (𝑓𝑡))
16 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = 𝑡𝑤 = 𝑡)
1715, 16eleq12d 2724 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 = 𝑡 → ((𝑓𝑤) ∈ 𝑤 ↔ (𝑓𝑡) ∈ 𝑡))
18 neeq2 2886 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = 𝑡 → (𝑧𝑤𝑧𝑡))
19 ineq2 3841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = 𝑡 → (𝑧𝑤) = (𝑧𝑡))
2019eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = 𝑡 → ((𝑧𝑤) = ∅ ↔ (𝑧𝑡) = ∅))
2118, 20imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 = 𝑡 → ((𝑧𝑤 → (𝑧𝑤) = ∅) ↔ (𝑧𝑡 → (𝑧𝑡) = ∅)))
2217, 21anbi12d 747 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 = 𝑡 → (((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) ↔ ((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅))))
2322rspcv 3336 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡𝑥 → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → ((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅))))
24 eleq1 2718 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑓𝑡) = 𝑣 → ((𝑓𝑡) ∈ 𝑧𝑣𝑧))
2524biimpar 501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓𝑡) = 𝑣𝑣𝑧) → (𝑓𝑡) ∈ 𝑧)
26 minel 4066 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡) = ∅) → ¬ (𝑓𝑡) ∈ 𝑧)
2726ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑓𝑡) ∈ 𝑡 → ((𝑧𝑡) = ∅ → ¬ (𝑓𝑡) ∈ 𝑧))
2827imim2d 57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓𝑡) ∈ 𝑡 → ((𝑧𝑡 → (𝑧𝑡) = ∅) → (𝑧𝑡 → ¬ (𝑓𝑡) ∈ 𝑧)))
2928imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) → (𝑧𝑡 → ¬ (𝑓𝑡) ∈ 𝑧))
3029necon4ad 2842 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) → ((𝑓𝑡) ∈ 𝑧𝑧 = 𝑡))
3130imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) ∧ (𝑓𝑡) ∈ 𝑧) → 𝑧 = 𝑡)
3225, 31sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) ∧ ((𝑓𝑡) = 𝑣𝑣𝑧)) → 𝑧 = 𝑡)
33 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = 𝑡 → (𝑓𝑧) = (𝑓𝑡))
34 eqeq2 2662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓𝑡) = 𝑣 → ((𝑓𝑧) = (𝑓𝑡) ↔ (𝑓𝑧) = 𝑣))
35 eqcom 2658 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓𝑧) = 𝑣𝑣 = (𝑓𝑧))
3634, 35syl6bb 276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑓𝑡) = 𝑣 → ((𝑓𝑧) = (𝑓𝑡) ↔ 𝑣 = (𝑓𝑧)))
3733, 36syl5ib 234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑓𝑡) = 𝑣 → (𝑧 = 𝑡𝑣 = (𝑓𝑧)))
3837ad2antrl 764 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) ∧ ((𝑓𝑡) = 𝑣𝑣𝑧)) → (𝑧 = 𝑡𝑣 = (𝑓𝑧)))
3932, 38mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) ∧ ((𝑓𝑡) = 𝑣𝑣𝑧)) → 𝑣 = (𝑓𝑧))
4039exp32 630 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) → ((𝑓𝑡) = 𝑣 → (𝑣𝑧𝑣 = (𝑓𝑧))))
4123, 40syl6com 37 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑡𝑥 → ((𝑓𝑡) = 𝑣 → (𝑣𝑧𝑣 = (𝑓𝑧)))))
4241com14 96 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣𝑧 → (𝑡𝑥 → ((𝑓𝑡) = 𝑣 → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → 𝑣 = (𝑓𝑧)))))
4342rexlimdv 3059 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣𝑧 → (∃𝑡𝑥 (𝑓𝑡) = 𝑣 → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → 𝑣 = (𝑓𝑧))))
4414, 43syl5 34 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣𝑧 → ((𝑣 ∈ ran 𝑓𝑓 Fn 𝑥) → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → 𝑣 = (𝑓𝑧))))
4544expd 451 . . . . . . . . . . . . . . . . . . . . 21 (𝑣𝑧 → (𝑣 ∈ ran 𝑓 → (𝑓 Fn 𝑥 → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → 𝑣 = (𝑓𝑧)))))
4645com4t 93 . . . . . . . . . . . . . . . . . . . 20 (𝑓 Fn 𝑥 → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑣𝑧 → (𝑣 ∈ ran 𝑓𝑣 = (𝑓𝑧)))))
4746imp4b 612 . . . . . . . . . . . . . . . . . . 19 ((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅))) → ((𝑣𝑧𝑣 ∈ ran 𝑓) → 𝑣 = (𝑓𝑧)))
4812, 47syl5bi 232 . . . . . . . . . . . . . . . . . 18 ((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅))) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓𝑧)))
4911, 48sylan2br 492 . . . . . . . . . . . . . . . . 17 ((𝑓 Fn 𝑥 ∧ (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅))) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓𝑧)))
5049anassrs 681 . . . . . . . . . . . . . . . 16 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓𝑧)))
5150adantlr 751 . . . . . . . . . . . . . . 15 ((((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓𝑧)))
52 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑧 → (𝑓𝑤) = (𝑓𝑧))
53 id 22 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑧𝑤 = 𝑧)
5452, 53eleq12d 2724 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑧 → ((𝑓𝑤) ∈ 𝑤 ↔ (𝑓𝑧) ∈ 𝑧))
5554rspcv 3336 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑥 → (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 → (𝑓𝑧) ∈ 𝑧))
56 fnfvelrn 6396 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓 Fn 𝑥𝑧𝑥) → (𝑓𝑧) ∈ ran 𝑓)
5756expcom 450 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑥 → (𝑓 Fn 𝑥 → (𝑓𝑧) ∈ ran 𝑓))
5855, 57anim12d 585 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑥 → ((∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤𝑓 Fn 𝑥) → ((𝑓𝑧) ∈ 𝑧 ∧ (𝑓𝑧) ∈ ran 𝑓)))
59 elin 3829 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓) ↔ ((𝑓𝑧) ∈ 𝑧 ∧ (𝑓𝑧) ∈ ran 𝑓))
6058, 59syl6ibr 242 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝑥 → ((∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤𝑓 Fn 𝑥) → (𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓)))
6160expd 451 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑥 → (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 → (𝑓 Fn 𝑥 → (𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓))))
6261com13 88 . . . . . . . . . . . . . . . . . 18 (𝑓 Fn 𝑥 → (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 → (𝑧𝑥 → (𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓))))
6362imp31 447 . . . . . . . . . . . . . . . . 17 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) → (𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓))
64 eleq1 2718 . . . . . . . . . . . . . . . . 17 (𝑣 = (𝑓𝑧) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ (𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓)))
6563, 64syl5ibrcom 237 . . . . . . . . . . . . . . . 16 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) → (𝑣 = (𝑓𝑧) → 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
6665adantr 480 . . . . . . . . . . . . . . 15 ((((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑣 = (𝑓𝑧) → 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
6751, 66impbid 202 . . . . . . . . . . . . . 14 ((((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧)))
6867ex 449 . . . . . . . . . . . . 13 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) → (∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧))))
6968alrimdv 1897 . . . . . . . . . . . 12 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) → (∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧))))
70 fvex 6239 . . . . . . . . . . . . . 14 (𝑓𝑧) ∈ V
71 eqeq2 2662 . . . . . . . . . . . . . . . 16 ( = (𝑓𝑧) → (𝑣 = 𝑣 = (𝑓𝑧)))
7271bibi2d 331 . . . . . . . . . . . . . . 15 ( = (𝑓𝑧) → ((𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ) ↔ (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧))))
7372albidv 1889 . . . . . . . . . . . . . 14 ( = (𝑓𝑧) → (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ) ↔ ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧))))
7470, 73spcev 3331 . . . . . . . . . . . . 13 (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧)) → ∃𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ))
75 df-eu 2502 . . . . . . . . . . . . 13 (∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ ∃𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ))
7674, 75sylibr 224 . . . . . . . . . . . 12 (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧)) → ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))
7769, 76syl6 35 . . . . . . . . . . 11 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) → (∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
7877ralimdva 2991 . . . . . . . . . 10 ((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) → (∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
7978ex 449 . . . . . . . . 9 (𝑓 Fn 𝑥 → (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 → (∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))))
8010, 79syl5 34 . . . . . . . 8 (𝑓 Fn 𝑥 → ((∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑧𝑥 𝑧 ≠ ∅) → (∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))))
8180expd 451 . . . . . . 7 (𝑓 Fn 𝑥 → (∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) → (∀𝑧𝑥 𝑧 ≠ ∅ → (∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))))
8281imp4b 612 . . . . . 6 ((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)) → ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
83 vex 3234 . . . . . . . 8 𝑓 ∈ V
8483rnex 7142 . . . . . . 7 ran 𝑓 ∈ V
85 ineq2 3841 . . . . . . . . . 10 (𝑦 = ran 𝑓 → (𝑧𝑦) = (𝑧 ∩ ran 𝑓))
8685eleq2d 2716 . . . . . . . . 9 (𝑦 = ran 𝑓 → (𝑣 ∈ (𝑧𝑦) ↔ 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
8786eubidv 2518 . . . . . . . 8 (𝑦 = ran 𝑓 → (∃!𝑣 𝑣 ∈ (𝑧𝑦) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
8887ralbidv 3015 . . . . . . 7 (𝑦 = ran 𝑓 → (∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦) ↔ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
8984, 88spcev 3331 . . . . . 6 (∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))
9082, 89syl6 35 . . . . 5 ((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)) → ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
9190exlimiv 1898 . . . 4 (∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)) → ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
9291alimi 1779 . . 3 (∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)) → ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
931, 92sylbi 207 . 2 (CHOICE → ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
94 eqid 2651 . . . . 5 {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))} = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))}
95 eqid 2651 . . . . 5 ( {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))} ∩ 𝑦) = ( {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))} ∩ 𝑦)
96 biid 251 . . . . 5 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
9794, 95, 96dfac5lem5 8988 . . . 4 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) → ∃𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
9897alrimiv 1895 . . 3 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) → ∀𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
99 dfac3 8982 . . 3 (CHOICE ↔ ∀𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
10098, 99sylibr 224 . 2 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) → CHOICE)
10193, 100impbii 199 1 (CHOICE ↔ ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  ∃!weu 2498  {cab 2637  wne 2823  wral 2941  wrex 2942  cin 3606  c0 3948  {csn 4210   cuni 4468   × cxp 5141  ran crn 5144   Fn wfn 5921  cfv 5926  CHOICEwac 8976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ac 8977
This theorem is referenced by:  dfackm  9026  ac8  9352
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