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Theorem dfac5 8812
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 8806 . . 3 (CHOICE ↔ ∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)))
2 neeq1 2843 . . . . . . . . . . . . 13 (𝑧 = 𝑤 → (𝑧 ≠ ∅ ↔ 𝑤 ≠ ∅))
32cbvralv 3146 . . . . . . . . . . . 12 (∀𝑧𝑥 𝑧 ≠ ∅ ↔ ∀𝑤𝑥 𝑤 ≠ ∅)
43anbi2i 725 . . . . . . . . . . 11 ((∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑧𝑥 𝑧 ≠ ∅) ↔ (∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑤𝑥 𝑤 ≠ ∅))
5 r19.26 3045 . . . . . . . . . . 11 (∀𝑤𝑥 ((𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) ↔ (∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑤𝑥 𝑤 ≠ ∅))
64, 5bitr4i 265 . . . . . . . . . 10 ((∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑧𝑥 𝑧 ≠ ∅) ↔ ∀𝑤𝑥 ((𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅))
7 pm3.35 608 . . . . . . . . . . . 12 ((𝑤 ≠ ∅ ∧ (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)) → (𝑓𝑤) ∈ 𝑤)
87ancoms 467 . . . . . . . . . . 11 (((𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) → (𝑓𝑤) ∈ 𝑤)
98ralimi 2935 . . . . . . . . . 10 (∀𝑤𝑥 ((𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) → ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤)
106, 9sylbi 205 . . . . . . . . 9 ((∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑧𝑥 𝑧 ≠ ∅) → ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤)
11 r19.26 3045 . . . . . . . . . . . . . . . . . 18 (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) ↔ (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)))
12 elin 3757 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ (𝑣𝑧𝑣 ∈ ran 𝑓))
13 fvelrnb 6138 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 Fn 𝑥 → (𝑣 ∈ ran 𝑓 ↔ ∃𝑡𝑥 (𝑓𝑡) = 𝑣))
1413biimpac 501 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑣 ∈ ran 𝑓𝑓 Fn 𝑥) → ∃𝑡𝑥 (𝑓𝑡) = 𝑣)
15 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = 𝑡 → (𝑓𝑤) = (𝑓𝑡))
16 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = 𝑡𝑤 = 𝑡)
1715, 16eleq12d 2681 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 = 𝑡 → ((𝑓𝑤) ∈ 𝑤 ↔ (𝑓𝑡) ∈ 𝑡))
18 neeq2 2844 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = 𝑡 → (𝑧𝑤𝑧𝑡))
19 ineq2 3769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = 𝑡 → (𝑧𝑤) = (𝑧𝑡))
2019eqeq1d 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = 𝑡 → ((𝑧𝑤) = ∅ ↔ (𝑧𝑡) = ∅))
2118, 20imbi12d 332 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 = 𝑡 → ((𝑧𝑤 → (𝑧𝑤) = ∅) ↔ (𝑧𝑡 → (𝑧𝑡) = ∅)))
2217, 21anbi12d 742 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 = 𝑡 → (((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) ↔ ((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅))))
2322rspcv 3277 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡𝑥 → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → ((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅))))
24 eleq1 2675 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑓𝑡) = 𝑣 → ((𝑓𝑡) ∈ 𝑧𝑣𝑧))
2524biimpar 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓𝑡) = 𝑣𝑣𝑧) → (𝑓𝑡) ∈ 𝑧)
26 minel 3984 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡) = ∅) → ¬ (𝑓𝑡) ∈ 𝑧)
2726ex 448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑓𝑡) ∈ 𝑡 → ((𝑧𝑡) = ∅ → ¬ (𝑓𝑡) ∈ 𝑧))
2827imim2d 54 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓𝑡) ∈ 𝑡 → ((𝑧𝑡 → (𝑧𝑡) = ∅) → (𝑧𝑡 → ¬ (𝑓𝑡) ∈ 𝑧)))
2928imp 443 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) → (𝑧𝑡 → ¬ (𝑓𝑡) ∈ 𝑧))
3029necon4ad 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) → ((𝑓𝑡) ∈ 𝑧𝑧 = 𝑡))
3130imp 443 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) ∧ (𝑓𝑡) ∈ 𝑧) → 𝑧 = 𝑡)
3225, 31sylan2 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) ∧ ((𝑓𝑡) = 𝑣𝑣𝑧)) → 𝑧 = 𝑡)
33 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = 𝑡 → (𝑓𝑧) = (𝑓𝑡))
34 eqeq2 2620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓𝑡) = 𝑣 → ((𝑓𝑧) = (𝑓𝑡) ↔ (𝑓𝑧) = 𝑣))
35 eqcom 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓𝑧) = 𝑣𝑣 = (𝑓𝑧))
3634, 35syl6bb 274 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑓𝑡) = 𝑣 → ((𝑓𝑧) = (𝑓𝑡) ↔ 𝑣 = (𝑓𝑧)))
3733, 36syl5ib 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑓𝑡) = 𝑣 → (𝑧 = 𝑡𝑣 = (𝑓𝑧)))
3837ad2antrl 759 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) ∧ ((𝑓𝑡) = 𝑣𝑣𝑧)) → (𝑧 = 𝑡𝑣 = (𝑓𝑧)))
3932, 38mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) ∧ ((𝑓𝑡) = 𝑣𝑣𝑧)) → 𝑣 = (𝑓𝑧))
4039exp32 628 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) → ((𝑓𝑡) = 𝑣 → (𝑣𝑧𝑣 = (𝑓𝑧))))
4123, 40syl6com 36 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑡𝑥 → ((𝑓𝑡) = 𝑣 → (𝑣𝑧𝑣 = (𝑓𝑧)))))
4241com14 93 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣𝑧 → (𝑡𝑥 → ((𝑓𝑡) = 𝑣 → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → 𝑣 = (𝑓𝑧)))))
4342rexlimdv 3011 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣𝑧 → (∃𝑡𝑥 (𝑓𝑡) = 𝑣 → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → 𝑣 = (𝑓𝑧))))
4414, 43syl5 33 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣𝑧 → ((𝑣 ∈ ran 𝑓𝑓 Fn 𝑥) → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → 𝑣 = (𝑓𝑧))))
4544expd 450 . . . . . . . . . . . . . . . . . . . . 21 (𝑣𝑧 → (𝑣 ∈ ran 𝑓 → (𝑓 Fn 𝑥 → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → 𝑣 = (𝑓𝑧)))))
4645com4t 90 . . . . . . . . . . . . . . . . . . . 20 (𝑓 Fn 𝑥 → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑣𝑧 → (𝑣 ∈ ran 𝑓𝑣 = (𝑓𝑧)))))
4746imp4b 610 . . . . . . . . . . . . . . . . . . 19 ((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅))) → ((𝑣𝑧𝑣 ∈ ran 𝑓) → 𝑣 = (𝑓𝑧)))
4812, 47syl5bi 230 . . . . . . . . . . . . . . . . . 18 ((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅))) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓𝑧)))
4911, 48sylan2br 491 . . . . . . . . . . . . . . . . 17 ((𝑓 Fn 𝑥 ∧ (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅))) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓𝑧)))
5049anassrs 677 . . . . . . . . . . . . . . . 16 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓𝑧)))
5150adantlr 746 . . . . . . . . . . . . . . 15 ((((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓𝑧)))
52 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑧 → (𝑓𝑤) = (𝑓𝑧))
53 id 22 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑧𝑤 = 𝑧)
5452, 53eleq12d 2681 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑧 → ((𝑓𝑤) ∈ 𝑤 ↔ (𝑓𝑧) ∈ 𝑧))
5554rspcv 3277 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑥 → (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 → (𝑓𝑧) ∈ 𝑧))
56 fnfvelrn 6249 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓 Fn 𝑥𝑧𝑥) → (𝑓𝑧) ∈ ran 𝑓)
5756expcom 449 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑥 → (𝑓 Fn 𝑥 → (𝑓𝑧) ∈ ran 𝑓))
5855, 57anim12d 583 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑥 → ((∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤𝑓 Fn 𝑥) → ((𝑓𝑧) ∈ 𝑧 ∧ (𝑓𝑧) ∈ ran 𝑓)))
59 elin 3757 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓) ↔ ((𝑓𝑧) ∈ 𝑧 ∧ (𝑓𝑧) ∈ ran 𝑓))
6058, 59syl6ibr 240 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝑥 → ((∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤𝑓 Fn 𝑥) → (𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓)))
6160expd 450 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑥 → (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 → (𝑓 Fn 𝑥 → (𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓))))
6261com13 85 . . . . . . . . . . . . . . . . . 18 (𝑓 Fn 𝑥 → (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 → (𝑧𝑥 → (𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓))))
6362imp31 446 . . . . . . . . . . . . . . . . 17 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) → (𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓))
64 eleq1 2675 . . . . . . . . . . . . . . . . 17 (𝑣 = (𝑓𝑧) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ (𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓)))
6563, 64syl5ibrcom 235 . . . . . . . . . . . . . . . 16 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) → (𝑣 = (𝑓𝑧) → 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
6665adantr 479 . . . . . . . . . . . . . . 15 ((((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑣 = (𝑓𝑧) → 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
6751, 66impbid 200 . . . . . . . . . . . . . 14 ((((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧)))
6867ex 448 . . . . . . . . . . . . 13 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) → (∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧))))
6968alrimdv 1843 . . . . . . . . . . . 12 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) → (∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧))))
70 fvex 6098 . . . . . . . . . . . . . 14 (𝑓𝑧) ∈ V
71 eqeq2 2620 . . . . . . . . . . . . . . . 16 ( = (𝑓𝑧) → (𝑣 = 𝑣 = (𝑓𝑧)))
7271bibi2d 330 . . . . . . . . . . . . . . 15 ( = (𝑓𝑧) → ((𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ) ↔ (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧))))
7372albidv 1835 . . . . . . . . . . . . . 14 ( = (𝑓𝑧) → (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ) ↔ ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧))))
7470, 73spcev 3272 . . . . . . . . . . . . 13 (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧)) → ∃𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ))
75 df-eu 2461 . . . . . . . . . . . . 13 (∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ ∃𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ))
7674, 75sylibr 222 . . . . . . . . . . . 12 (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧)) → ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))
7769, 76syl6 34 . . . . . . . . . . 11 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) → (∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
7877ralimdva 2944 . . . . . . . . . 10 ((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) → (∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
7978ex 448 . . . . . . . . 9 (𝑓 Fn 𝑥 → (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 → (∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))))
8010, 79syl5 33 . . . . . . . 8 (𝑓 Fn 𝑥 → ((∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑧𝑥 𝑧 ≠ ∅) → (∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))))
8180expd 450 . . . . . . 7 (𝑓 Fn 𝑥 → (∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) → (∀𝑧𝑥 𝑧 ≠ ∅ → (∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))))
8281imp4b 610 . . . . . 6 ((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)) → ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
83 vex 3175 . . . . . . . 8 𝑓 ∈ V
8483rnex 6970 . . . . . . 7 ran 𝑓 ∈ V
85 ineq2 3769 . . . . . . . . . 10 (𝑦 = ran 𝑓 → (𝑧𝑦) = (𝑧 ∩ ran 𝑓))
8685eleq2d 2672 . . . . . . . . 9 (𝑦 = ran 𝑓 → (𝑣 ∈ (𝑧𝑦) ↔ 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
8786eubidv 2477 . . . . . . . 8 (𝑦 = ran 𝑓 → (∃!𝑣 𝑣 ∈ (𝑧𝑦) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
8887ralbidv 2968 . . . . . . 7 (𝑦 = ran 𝑓 → (∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦) ↔ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
8984, 88spcev 3272 . . . . . 6 (∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))
9082, 89syl6 34 . . . . 5 ((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)) → ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
9190exlimiv 1844 . . . 4 (∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)) → ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
9291alimi 1729 . . 3 (∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)) → ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
931, 92sylbi 205 . 2 (CHOICE → ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
94 eqid 2609 . . . . 5 {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))} = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))}
95 eqid 2609 . . . . 5 ( {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))} ∩ 𝑦) = ( {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))} ∩ 𝑦)
96 biid 249 . . . . 5 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
9794, 95, 96dfac5lem5 8811 . . . 4 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) → ∃𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
9897alrimiv 1841 . . 3 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) → ∀𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
99 dfac3 8805 . . 3 (CHOICE ↔ ∀𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
10098, 99sylibr 222 . 2 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) → CHOICE)
10193, 100impbii 197 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 194  wa 382  wal 1472   = wceq 1474  wex 1694  wcel 1976  ∃!weu 2457  {cab 2595  wne 2779  wral 2895  wrex 2896  cin 3538  c0 3873  {csn 4124   cuni 4366   × cxp 5026  ran crn 5029   Fn wfn 5785  cfv 5790  CHOICEwac 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-ac 8800
This theorem is referenced by:  dfackm  8849  ac8  9175
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