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Theorem dfac2 8836
 Description: Axiom of Choice (first form) of [Enderton] p. 49 implies of our Axiom of Choice (in the form of ac3 9167). The proof does not make use of AC. Note that the Axiom of Regularity is used by the proof. Specifically, elirrv 8387 and preleq 8397 that are referenced in the proof each make use of Regularity for their derivations. (The reverse implication can be derived without using Regularity; see dfac2a 8835.) TODO: Fix label in comment, and put label changes into list at top of set.mm. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
dfac2 (CHOICE ↔ ∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣

Proof of Theorem dfac2
Dummy variables 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfac3 8827 . . 3 (CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
2 nfra1 2925 . . . . . . 7 𝑧𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
3 rsp 2913 . . . . . . . . . . . . 13 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑧𝑥 → (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
4 equid 1926 . . . . . . . . . . . . . . . . . . 19 𝑧 = 𝑧
5 neeq1 2844 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑧 → (𝑢 ≠ ∅ ↔ 𝑧 ≠ ∅))
6 eqeq1 2614 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑧 → (𝑢 = 𝑧𝑧 = 𝑧))
75, 6anbi12d 743 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑧 → ((𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) ↔ (𝑧 ≠ ∅ ∧ 𝑧 = 𝑧)))
87rspcev 3282 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝑥 ∧ (𝑧 ≠ ∅ ∧ 𝑧 = 𝑧)) → ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧))
94, 8mpanr2 716 . . . . . . . . . . . . . . . . . 18 ((𝑧𝑥𝑧 ≠ ∅) → ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧))
10 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑧 → (𝑓𝑢) = (𝑓𝑧))
1110preq1d 4218 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑧 → {(𝑓𝑢), 𝑢} = {(𝑓𝑧), 𝑢})
12 preq2 4213 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑧 → {(𝑓𝑧), 𝑢} = {(𝑓𝑧), 𝑧})
1311, 12eqtr2d 2645 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑧 → {(𝑓𝑧), 𝑧} = {(𝑓𝑢), 𝑢})
1413anim2i 591 . . . . . . . . . . . . . . . . . . 19 ((𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) → (𝑢 ≠ ∅ ∧ {(𝑓𝑧), 𝑧} = {(𝑓𝑢), 𝑢}))
1514reximi 2994 . . . . . . . . . . . . . . . . . 18 (∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) → ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ {(𝑓𝑧), 𝑧} = {(𝑓𝑢), 𝑢}))
169, 15syl 17 . . . . . . . . . . . . . . . . 17 ((𝑧𝑥𝑧 ≠ ∅) → ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ {(𝑓𝑧), 𝑧} = {(𝑓𝑢), 𝑢}))
17 prex 4836 . . . . . . . . . . . . . . . . . 18 {(𝑓𝑧), 𝑧} ∈ V
18 eqeq1 2614 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = {(𝑓𝑧), 𝑧} → (𝑔 = {(𝑓𝑢), 𝑢} ↔ {(𝑓𝑧), 𝑧} = {(𝑓𝑢), 𝑢}))
1918anbi2d 736 . . . . . . . . . . . . . . . . . . 19 (𝑔 = {(𝑓𝑧), 𝑧} → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢}) ↔ (𝑢 ≠ ∅ ∧ {(𝑓𝑧), 𝑧} = {(𝑓𝑢), 𝑢})))
2019rexbidv 3034 . . . . . . . . . . . . . . . . . 18 (𝑔 = {(𝑓𝑧), 𝑧} → (∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢}) ↔ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ {(𝑓𝑧), 𝑧} = {(𝑓𝑢), 𝑢})))
2117, 20elab 3319 . . . . . . . . . . . . . . . . 17 ({(𝑓𝑧), 𝑧} ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} ↔ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ {(𝑓𝑧), 𝑧} = {(𝑓𝑢), 𝑢}))
2216, 21sylibr 223 . . . . . . . . . . . . . . . 16 ((𝑧𝑥𝑧 ≠ ∅) → {(𝑓𝑧), 𝑧} ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})})
23 vex 3176 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
2423prid2 4242 . . . . . . . . . . . . . . . . 17 𝑧 ∈ {(𝑓𝑧), 𝑧}
25 fvex 6113 . . . . . . . . . . . . . . . . . 18 (𝑓𝑧) ∈ V
2625prid1 4241 . . . . . . . . . . . . . . . . 17 (𝑓𝑧) ∈ {(𝑓𝑧), 𝑧}
2724, 26pm3.2i 470 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {(𝑓𝑧), 𝑧} ∧ (𝑓𝑧) ∈ {(𝑓𝑧), 𝑧})
28 eleq2 2677 . . . . . . . . . . . . . . . . . 18 (𝑣 = {(𝑓𝑧), 𝑧} → (𝑧𝑣𝑧 ∈ {(𝑓𝑧), 𝑧}))
29 eleq2 2677 . . . . . . . . . . . . . . . . . 18 (𝑣 = {(𝑓𝑧), 𝑧} → ((𝑓𝑧) ∈ 𝑣 ↔ (𝑓𝑧) ∈ {(𝑓𝑧), 𝑧}))
3028, 29anbi12d 743 . . . . . . . . . . . . . . . . 17 (𝑣 = {(𝑓𝑧), 𝑧} → ((𝑧𝑣 ∧ (𝑓𝑧) ∈ 𝑣) ↔ (𝑧 ∈ {(𝑓𝑧), 𝑧} ∧ (𝑓𝑧) ∈ {(𝑓𝑧), 𝑧})))
3130rspcev 3282 . . . . . . . . . . . . . . . 16 (({(𝑓𝑧), 𝑧} ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} ∧ (𝑧 ∈ {(𝑓𝑧), 𝑧} ∧ (𝑓𝑧) ∈ {(𝑓𝑧), 𝑧})) → ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣 ∧ (𝑓𝑧) ∈ 𝑣))
3222, 27, 31sylancl 693 . . . . . . . . . . . . . . 15 ((𝑧𝑥𝑧 ≠ ∅) → ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣 ∧ (𝑓𝑧) ∈ 𝑣))
33 eleq1 2676 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑓𝑧) → (𝑤𝑧 ↔ (𝑓𝑧) ∈ 𝑧))
34 eleq1 2676 . . . . . . . . . . . . . . . . . . 19 (𝑤 = (𝑓𝑧) → (𝑤𝑣 ↔ (𝑓𝑧) ∈ 𝑣))
3534anbi2d 736 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑓𝑧) → ((𝑧𝑣𝑤𝑣) ↔ (𝑧𝑣 ∧ (𝑓𝑧) ∈ 𝑣)))
3635rexbidv 3034 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑓𝑧) → (∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣) ↔ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣 ∧ (𝑓𝑧) ∈ 𝑣)))
3733, 36anbi12d 743 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑧) → ((𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)) ↔ ((𝑓𝑧) ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣 ∧ (𝑓𝑧) ∈ 𝑣))))
3825, 37spcev 3273 . . . . . . . . . . . . . . 15 (((𝑓𝑧) ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣 ∧ (𝑓𝑧) ∈ 𝑣)) → ∃𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
3932, 38sylan2 490 . . . . . . . . . . . . . 14 (((𝑓𝑧) ∈ 𝑧 ∧ (𝑧𝑥𝑧 ≠ ∅)) → ∃𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
4039ex 449 . . . . . . . . . . . . 13 ((𝑓𝑧) ∈ 𝑧 → ((𝑧𝑥𝑧 ≠ ∅) → ∃𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣))))
413, 40syl8 74 . . . . . . . . . . . 12 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑧𝑥 → (𝑧 ≠ ∅ → ((𝑧𝑥𝑧 ≠ ∅) → ∃𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣))))))
4241impd 446 . . . . . . . . . . 11 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝑧𝑥𝑧 ≠ ∅) → ((𝑧𝑥𝑧 ≠ ∅) → ∃𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))))
4342pm2.43d 51 . . . . . . . . . 10 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝑧𝑥𝑧 ≠ ∅) → ∃𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣))))
44 df-rex 2902 . . . . . . . . . . . . . 14 (∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣) ↔ ∃𝑣(𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} ∧ (𝑧𝑣𝑤𝑣)))
45 vex 3176 . . . . . . . . . . . . . . . . . . . 20 𝑣 ∈ V
46 eqeq1 2614 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔 = 𝑣 → (𝑔 = {(𝑓𝑢), 𝑢} ↔ 𝑣 = {(𝑓𝑢), 𝑢}))
4746anbi2d 736 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑣 → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢}) ↔ (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓𝑢), 𝑢})))
4847rexbidv 3034 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑣 → (∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢}) ↔ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓𝑢), 𝑢})))
4945, 48elab 3319 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} ↔ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓𝑢), 𝑢}))
50 neeq1 2844 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑢 → (𝑧 ≠ ∅ ↔ 𝑢 ≠ ∅))
51 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = 𝑢 → (𝑓𝑧) = (𝑓𝑢))
5251eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑢 → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓𝑢) ∈ 𝑧))
53 eleq2 2677 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑢 → ((𝑓𝑢) ∈ 𝑧 ↔ (𝑓𝑢) ∈ 𝑢))
5452, 53bitrd 267 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑢 → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓𝑢) ∈ 𝑢))
5550, 54imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = 𝑢 → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑢 ≠ ∅ → (𝑓𝑢) ∈ 𝑢)))
5655rspccv 3279 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑢𝑥 → (𝑢 ≠ ∅ → (𝑓𝑢) ∈ 𝑢)))
57 elirrv 8387 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ¬ 𝑤𝑤
58 eleq2 2677 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑤 = 𝑧 → (𝑤𝑤𝑤𝑧))
5957, 58mtbii 315 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑤 = 𝑧 → ¬ 𝑤𝑧)
6059con2i 133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑤𝑧 → ¬ 𝑤 = 𝑧)
61 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑤 ∈ V
62 fvex 6113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑓𝑢) ∈ V
63 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑢 ∈ V
6461, 23, 62, 63prel12 4323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑤 = 𝑧 → ({𝑤, 𝑧} = {(𝑓𝑢), 𝑢} ↔ (𝑤 ∈ {(𝑓𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓𝑢), 𝑢})))
65 ancom 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑤𝑣𝑧𝑣) ↔ (𝑧𝑣𝑤𝑣))
66 eleq2 2677 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑣 = {(𝑓𝑢), 𝑢} → (𝑤𝑣𝑤 ∈ {(𝑓𝑢), 𝑢}))
67 eleq2 2677 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑣 = {(𝑓𝑢), 𝑢} → (𝑧𝑣𝑧 ∈ {(𝑓𝑢), 𝑢}))
6866, 67anbi12d 743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑣 = {(𝑓𝑢), 𝑢} → ((𝑤𝑣𝑧𝑣) ↔ (𝑤 ∈ {(𝑓𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓𝑢), 𝑢})))
6965, 68syl5rbbr 274 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑣 = {(𝑓𝑢), 𝑢} → ((𝑤 ∈ {(𝑓𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓𝑢), 𝑢}) ↔ (𝑧𝑣𝑤𝑣)))
7064, 69sylan9bbr 733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑣 = {(𝑓𝑢), 𝑢} ∧ ¬ 𝑤 = 𝑧) → ({𝑤, 𝑧} = {(𝑓𝑢), 𝑢} ↔ (𝑧𝑣𝑤𝑣)))
7160, 70sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑣 = {(𝑓𝑢), 𝑢} ∧ 𝑤𝑧) → ({𝑤, 𝑧} = {(𝑓𝑢), 𝑢} ↔ (𝑧𝑣𝑤𝑣)))
7271adantrr 749 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑣 = {(𝑓𝑢), 𝑢} ∧ (𝑤𝑧 ∧ (𝑓𝑢) ∈ 𝑢)) → ({𝑤, 𝑧} = {(𝑓𝑢), 𝑢} ↔ (𝑧𝑣𝑤𝑣)))
7372pm5.32da 671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑣 = {(𝑓𝑢), 𝑢} → (((𝑤𝑧 ∧ (𝑓𝑢) ∈ 𝑢) ∧ {𝑤, 𝑧} = {(𝑓𝑢), 𝑢}) ↔ ((𝑤𝑧 ∧ (𝑓𝑢) ∈ 𝑢) ∧ (𝑧𝑣𝑤𝑣))))
7461, 23, 62, 63preleq 8397 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑤𝑧 ∧ (𝑓𝑢) ∈ 𝑢) ∧ {𝑤, 𝑧} = {(𝑓𝑢), 𝑢}) → (𝑤 = (𝑓𝑢) ∧ 𝑧 = 𝑢))
7573, 74syl6bir 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑣 = {(𝑓𝑢), 𝑢} → (((𝑤𝑧 ∧ (𝑓𝑢) ∈ 𝑢) ∧ (𝑧𝑣𝑤𝑣)) → (𝑤 = (𝑓𝑢) ∧ 𝑧 = 𝑢)))
7651eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = 𝑢 → (𝑤 = (𝑓𝑧) ↔ 𝑤 = (𝑓𝑢)))
7776biimparc 503 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑤 = (𝑓𝑢) ∧ 𝑧 = 𝑢) → 𝑤 = (𝑓𝑧))
7875, 77syl6 34 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑣 = {(𝑓𝑢), 𝑢} → (((𝑤𝑧 ∧ (𝑓𝑢) ∈ 𝑢) ∧ (𝑧𝑣𝑤𝑣)) → 𝑤 = (𝑓𝑧)))
7978exp4c 634 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑣 = {(𝑓𝑢), 𝑢} → (𝑤𝑧 → ((𝑓𝑢) ∈ 𝑢 → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))))
8079com13 86 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓𝑢) ∈ 𝑢 → (𝑤𝑧 → (𝑣 = {(𝑓𝑢), 𝑢} → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))))
8156, 80syl8 74 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑢𝑥 → (𝑢 ≠ ∅ → (𝑤𝑧 → (𝑣 = {(𝑓𝑢), 𝑢} → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))))))
8281com4r 92 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤𝑧 → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑢𝑥 → (𝑢 ≠ ∅ → (𝑣 = {(𝑓𝑢), 𝑢} → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))))))
8382imp 444 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑤𝑧 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑢𝑥 → (𝑢 ≠ ∅ → (𝑣 = {(𝑓𝑢), 𝑢} → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧))))))
8483imp4a 612 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤𝑧 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑢𝑥 → ((𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓𝑢), 𝑢}) → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))))
8584com3l 87 . . . . . . . . . . . . . . . . . . . 20 (𝑢𝑥 → ((𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓𝑢), 𝑢}) → ((𝑤𝑧 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))))
8685rexlimiv 3009 . . . . . . . . . . . . . . . . . . 19 (∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓𝑢), 𝑢}) → ((𝑤𝑧 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧))))
8749, 86sylbi 206 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} → ((𝑤𝑧 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧))))
8887expd 451 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} → (𝑤𝑧 → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))))
8988com13 86 . . . . . . . . . . . . . . . 16 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑤𝑧 → (𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))))
9089imp4b 611 . . . . . . . . . . . . . . 15 ((∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤𝑧) → ((𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} ∧ (𝑧𝑣𝑤𝑣)) → 𝑤 = (𝑓𝑧)))
9190exlimdv 1848 . . . . . . . . . . . . . 14 ((∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤𝑧) → (∃𝑣(𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} ∧ (𝑧𝑣𝑤𝑣)) → 𝑤 = (𝑓𝑧)))
9244, 91syl5bi 231 . . . . . . . . . . . . 13 ((∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤𝑧) → (∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))
9392expimpd 627 . . . . . . . . . . . 12 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)) → 𝑤 = (𝑓𝑧)))
9493alrimiv 1842 . . . . . . . . . . 11 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∀𝑤((𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)) → 𝑤 = (𝑓𝑧)))
95 mo2icl 3352 . . . . . . . . . . 11 (∀𝑤((𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)) → 𝑤 = (𝑓𝑧)) → ∃*𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
9694, 95syl 17 . . . . . . . . . 10 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃*𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
9743, 96jctird 565 . . . . . . . . 9 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝑧𝑥𝑧 ≠ ∅) → (∃𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)) ∧ ∃*𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))))
98 df-reu 2903 . . . . . . . . . 10 (∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣) ↔ ∃!𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
99 eu5 2484 . . . . . . . . . 10 (∃!𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)) ↔ (∃𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)) ∧ ∃*𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣))))
10098, 99bitri 263 . . . . . . . . 9 (∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣) ↔ (∃𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)) ∧ ∃*𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣))))
10197, 100syl6ibr 241 . . . . . . . 8 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝑧𝑥𝑧 ≠ ∅) → ∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
102101expd 451 . . . . . . 7 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑧𝑥 → (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣))))
1032, 102ralrimi 2940 . . . . . 6 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
104 vex 3176 . . . . . . . . . . . 12 𝑓 ∈ V
105104rnex 6992 . . . . . . . . . . 11 ran 𝑓 ∈ V
106 p0ex 4779 . . . . . . . . . . 11 {∅} ∈ V
107105, 106unex 6854 . . . . . . . . . 10 (ran 𝑓 ∪ {∅}) ∈ V
108 vex 3176 . . . . . . . . . 10 𝑥 ∈ V
109107, 108unex 6854 . . . . . . . . 9 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∈ V
110109pwex 4774 . . . . . . . 8 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∈ V
111 ssun1 3738 . . . . . . . . . . . . . . 15 (ran 𝑓 ∪ {∅}) ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
112 fvrn0 6126 . . . . . . . . . . . . . . 15 (𝑓𝑢) ∈ (ran 𝑓 ∪ {∅})
113111, 112sselii 3565 . . . . . . . . . . . . . 14 (𝑓𝑢) ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
114 elun2 3743 . . . . . . . . . . . . . 14 (𝑢𝑥𝑢 ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
115 prssi 4293 . . . . . . . . . . . . . 14 (((𝑓𝑢) ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∧ 𝑢 ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)) → {(𝑓𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
116113, 114, 115sylancr 694 . . . . . . . . . . . . 13 (𝑢𝑥 → {(𝑓𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
117 prex 4836 . . . . . . . . . . . . . 14 {(𝑓𝑢), 𝑢} ∈ V
118117elpw 4114 . . . . . . . . . . . . 13 ({(𝑓𝑢), 𝑢} ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ↔ {(𝑓𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
119116, 118sylibr 223 . . . . . . . . . . . 12 (𝑢𝑥 → {(𝑓𝑢), 𝑢} ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
120 eleq1 2676 . . . . . . . . . . . 12 (𝑔 = {(𝑓𝑢), 𝑢} → (𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ↔ {(𝑓𝑢), 𝑢} ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)))
121119, 120syl5ibrcom 236 . . . . . . . . . . 11 (𝑢𝑥 → (𝑔 = {(𝑓𝑢), 𝑢} → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)))
122121adantld 482 . . . . . . . . . 10 (𝑢𝑥 → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢}) → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)))
123122rexlimiv 3009 . . . . . . . . 9 (∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢}) → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
124123abssi 3640 . . . . . . . 8 {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} ⊆ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
125110, 124ssexi 4731 . . . . . . 7 {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} ∈ V
126 rexeq 3116 . . . . . . . . . 10 (𝑦 = {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} → (∃𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
127126reubidv 3103 . . . . . . . . 9 (𝑦 = {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} → (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
128127imbi2d 329 . . . . . . . 8 (𝑦 = {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} → ((𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) ↔ (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣))))
129128ralbidv 2969 . . . . . . 7 (𝑦 = {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} → (∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣))))
130125, 129spcev 3273 . . . . . 6 (∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)) → ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
131103, 130syl 17 . . . . 5 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
132131exlimiv 1845 . . . 4 (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
133132alimi 1730 . . 3 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
1341, 133sylbi 206 . 2 (CHOICE → ∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
135 dfac2a 8835 . 2 (∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → CHOICE)
136134, 135impbii 198 1 (CHOICE ↔ ∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458  ∃*wmo 2459  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127  ran crn 5039  ‘cfv 5804  CHOICEwac 8821 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-eprel 4949  df-id 4953  df-fr 4997  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-riota 6511  df-ac 8822 This theorem is referenced by:  dfac7  8837
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