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Theorem dfac2b 9545
 Description: Axiom of Choice (first form) of [Enderton] p. 49 implies our Axiom of Choice (in the form of ac3 9873). The proof does not make use of AC. Note that the Axiom of Regularity is used by the proof. Specifically, elneq 9050 and preleq 9067 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 9544.) (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by AV, 16-Jun-2022.)
Assertion
Ref Expression
dfac2b (CHOICE → ∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
Distinct variable group:   𝑥,𝑧,𝑦,𝑤,𝑣

Proof of Theorem dfac2b
Dummy variables 𝑓 𝑢 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfac3 9536 . 2 (CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
2 nfra1 3208 . . . . . 6 𝑧𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
3 rsp 3195 . . . . . . . . . . . 12 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑧𝑥 → (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
4 equid 2019 . . . . . . . . . . . . . . . . . 18 𝑧 = 𝑧
5 neeq1 3073 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑧 → (𝑢 ≠ ∅ ↔ 𝑧 ≠ ∅))
6 eqeq1 2826 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑧 → (𝑢 = 𝑧𝑧 = 𝑧))
75, 6anbi12d 633 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑧 → ((𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) ↔ (𝑧 ≠ ∅ ∧ 𝑧 = 𝑧)))
87rspcev 3598 . . . . . . . . . . . . . . . . . 18 ((𝑧𝑥 ∧ (𝑧 ≠ ∅ ∧ 𝑧 = 𝑧)) → ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧))
94, 8mpanr2 703 . . . . . . . . . . . . . . . . 17 ((𝑧𝑥𝑧 ≠ ∅) → ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧))
10 fveq2 6652 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑧 → (𝑓𝑢) = (𝑓𝑧))
1110preq1d 4649 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑧 → {(𝑓𝑢), 𝑢} = {(𝑓𝑧), 𝑢})
12 preq2 4644 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑧 → {(𝑓𝑧), 𝑢} = {(𝑓𝑧), 𝑧})
1311, 12eqtr2d 2858 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑧 → {(𝑓𝑧), 𝑧} = {(𝑓𝑢), 𝑢})
1413anim2i 619 . . . . . . . . . . . . . . . . . 18 ((𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) → (𝑢 ≠ ∅ ∧ {(𝑓𝑧), 𝑧} = {(𝑓𝑢), 𝑢}))
1514reximi 3231 . . . . . . . . . . . . . . . . 17 (∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) → ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ {(𝑓𝑧), 𝑧} = {(𝑓𝑢), 𝑢}))
169, 15syl 17 . . . . . . . . . . . . . . . 16 ((𝑧𝑥𝑧 ≠ ∅) → ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ {(𝑓𝑧), 𝑧} = {(𝑓𝑢), 𝑢}))
17 prex 5310 . . . . . . . . . . . . . . . . 17 {(𝑓𝑧), 𝑧} ∈ V
18 eqeq1 2826 . . . . . . . . . . . . . . . . . . 19 (𝑔 = {(𝑓𝑧), 𝑧} → (𝑔 = {(𝑓𝑢), 𝑢} ↔ {(𝑓𝑧), 𝑧} = {(𝑓𝑢), 𝑢}))
1918anbi2d 631 . . . . . . . . . . . . . . . . . 18 (𝑔 = {(𝑓𝑧), 𝑧} → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢}) ↔ (𝑢 ≠ ∅ ∧ {(𝑓𝑧), 𝑧} = {(𝑓𝑢), 𝑢})))
2019rexbidv 3283 . . . . . . . . . . . . . . . . 17 (𝑔 = {(𝑓𝑧), 𝑧} → (∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢}) ↔ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ {(𝑓𝑧), 𝑧} = {(𝑓𝑢), 𝑢})))
2117, 20elab 3642 . . . . . . . . . . . . . . . 16 ({(𝑓𝑧), 𝑧} ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} ↔ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ {(𝑓𝑧), 𝑧} = {(𝑓𝑢), 𝑢}))
2216, 21sylibr 237 . . . . . . . . . . . . . . 15 ((𝑧𝑥𝑧 ≠ ∅) → {(𝑓𝑧), 𝑧} ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})})
23 vex 3472 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
2423prid2 4673 . . . . . . . . . . . . . . . 16 𝑧 ∈ {(𝑓𝑧), 𝑧}
25 fvex 6665 . . . . . . . . . . . . . . . . 17 (𝑓𝑧) ∈ V
2625prid1 4672 . . . . . . . . . . . . . . . 16 (𝑓𝑧) ∈ {(𝑓𝑧), 𝑧}
2724, 26pm3.2i 474 . . . . . . . . . . . . . . 15 (𝑧 ∈ {(𝑓𝑧), 𝑧} ∧ (𝑓𝑧) ∈ {(𝑓𝑧), 𝑧})
28 eleq2 2902 . . . . . . . . . . . . . . . . 17 (𝑣 = {(𝑓𝑧), 𝑧} → (𝑧𝑣𝑧 ∈ {(𝑓𝑧), 𝑧}))
29 eleq2 2902 . . . . . . . . . . . . . . . . 17 (𝑣 = {(𝑓𝑧), 𝑧} → ((𝑓𝑧) ∈ 𝑣 ↔ (𝑓𝑧) ∈ {(𝑓𝑧), 𝑧}))
3028, 29anbi12d 633 . . . . . . . . . . . . . . . 16 (𝑣 = {(𝑓𝑧), 𝑧} → ((𝑧𝑣 ∧ (𝑓𝑧) ∈ 𝑣) ↔ (𝑧 ∈ {(𝑓𝑧), 𝑧} ∧ (𝑓𝑧) ∈ {(𝑓𝑧), 𝑧})))
3130rspcev 3598 . . . . . . . . . . . . . . 15 (({(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 589 . . . . . . . . . . . . . 14 ((𝑧𝑥𝑧 ≠ ∅) → ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣 ∧ (𝑓𝑧) ∈ 𝑣))
33 eleq1 2901 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑧) → (𝑤𝑧 ↔ (𝑓𝑧) ∈ 𝑧))
34 eleq1 2901 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑓𝑧) → (𝑤𝑣 ↔ (𝑓𝑧) ∈ 𝑣))
3534anbi2d 631 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑓𝑧) → ((𝑧𝑣𝑤𝑣) ↔ (𝑧𝑣 ∧ (𝑓𝑧) ∈ 𝑣)))
3635rexbidv 3283 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑧) → (∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣) ↔ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣 ∧ (𝑓𝑧) ∈ 𝑣)))
3733, 36anbi12d 633 . . . . . . . . . . . . . . 15 (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 3582 . . . . . . . . . . . . . 14 (((𝑓𝑧) ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣 ∧ (𝑓𝑧) ∈ 𝑣)) → ∃𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
3932, 38sylan2 595 . . . . . . . . . . . . 13 (((𝑓𝑧) ∈ 𝑧 ∧ (𝑧𝑥𝑧 ≠ ∅)) → ∃𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
4039ex 416 . . . . . . . . . . . 12 ((𝑓𝑧) ∈ 𝑧 → ((𝑧𝑥𝑧 ≠ ∅) → ∃𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣))))
413, 40syl8 76 . . . . . . . . . . 11 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑧𝑥 → (𝑧 ≠ ∅ → ((𝑧𝑥𝑧 ≠ ∅) → ∃𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣))))))
4241impd 414 . . . . . . . . . 10 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝑧𝑥𝑧 ≠ ∅) → ((𝑧𝑥𝑧 ≠ ∅) → ∃𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))))
4342pm2.43d 53 . . . . . . . . 9 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝑧𝑥𝑧 ≠ ∅) → ∃𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣))))
44 df-rex 3136 . . . . . . . . . . . . 13 (∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣) ↔ ∃𝑣(𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} ∧ (𝑧𝑣𝑤𝑣)))
45 vex 3472 . . . . . . . . . . . . . . . . . . 19 𝑣 ∈ V
46 eqeq1 2826 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑣 → (𝑔 = {(𝑓𝑢), 𝑢} ↔ 𝑣 = {(𝑓𝑢), 𝑢}))
4746anbi2d 631 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑣 → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢}) ↔ (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓𝑢), 𝑢})))
4847rexbidv 3283 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝑣 → (∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢}) ↔ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓𝑢), 𝑢})))
4945, 48elab 3642 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} ↔ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓𝑢), 𝑢}))
50 neeq1 3073 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = 𝑢 → (𝑧 ≠ ∅ ↔ 𝑢 ≠ ∅))
51 fveq2 6652 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑢 → (𝑓𝑧) = (𝑓𝑢))
5251eleq1d 2898 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑢 → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓𝑢) ∈ 𝑧))
53 eleq2 2902 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑢 → ((𝑓𝑢) ∈ 𝑧 ↔ (𝑓𝑢) ∈ 𝑢))
5452, 53bitrd 282 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = 𝑢 → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓𝑢) ∈ 𝑢))
5550, 54imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑢 → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑢 ≠ ∅ → (𝑓𝑢) ∈ 𝑢)))
5655rspccv 3595 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑢𝑥 → (𝑢 ≠ ∅ → (𝑓𝑢) ∈ 𝑢)))
57 elneq 9050 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑤𝑧𝑤𝑧)
5857neneqd 3016 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤𝑧 → ¬ 𝑤 = 𝑧)
59 vex 3472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑤 ∈ V
60 neqne 3019 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑤 = 𝑧𝑤𝑧)
61 prel12g 4767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑤 ∈ V ∧ 𝑧 ∈ V ∧ 𝑤𝑧) → ({𝑤, 𝑧} = {(𝑓𝑢), 𝑢} ↔ (𝑤 ∈ {(𝑓𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓𝑢), 𝑢})))
6259, 23, 60, 61mp3an12i 1462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑤 = 𝑧 → ({𝑤, 𝑧} = {(𝑓𝑢), 𝑢} ↔ (𝑤 ∈ {(𝑓𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓𝑢), 𝑢})))
63 eleq2 2902 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑣 = {(𝑓𝑢), 𝑢} → (𝑤𝑣𝑤 ∈ {(𝑓𝑢), 𝑢}))
64 eleq2 2902 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑣 = {(𝑓𝑢), 𝑢} → (𝑧𝑣𝑧 ∈ {(𝑓𝑢), 𝑢}))
6563, 64anbi12d 633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑣 = {(𝑓𝑢), 𝑢} → ((𝑤𝑣𝑧𝑣) ↔ (𝑤 ∈ {(𝑓𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓𝑢), 𝑢})))
66 ancom 464 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑤𝑣𝑧𝑣) ↔ (𝑧𝑣𝑤𝑣))
6765, 66bitr3di 289 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑣 = {(𝑓𝑢), 𝑢} → ((𝑤 ∈ {(𝑓𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓𝑢), 𝑢}) ↔ (𝑧𝑣𝑤𝑣)))
6862, 67sylan9bbr 514 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑣 = {(𝑓𝑢), 𝑢} ∧ ¬ 𝑤 = 𝑧) → ({𝑤, 𝑧} = {(𝑓𝑢), 𝑢} ↔ (𝑧𝑣𝑤𝑣)))
6958, 68sylan2 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑣 = {(𝑓𝑢), 𝑢} ∧ 𝑤𝑧) → ({𝑤, 𝑧} = {(𝑓𝑢), 𝑢} ↔ (𝑧𝑣𝑤𝑣)))
7069adantrr 716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑣 = {(𝑓𝑢), 𝑢} ∧ (𝑤𝑧 ∧ (𝑓𝑢) ∈ 𝑢)) → ({𝑤, 𝑧} = {(𝑓𝑢), 𝑢} ↔ (𝑧𝑣𝑤𝑣)))
7170pm5.32da 582 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑣 = {(𝑓𝑢), 𝑢} → (((𝑤𝑧 ∧ (𝑓𝑢) ∈ 𝑢) ∧ {𝑤, 𝑧} = {(𝑓𝑢), 𝑢}) ↔ ((𝑤𝑧 ∧ (𝑓𝑢) ∈ 𝑢) ∧ (𝑧𝑣𝑤𝑣))))
7223preleq 9067 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑤𝑧 ∧ (𝑓𝑢) ∈ 𝑢) ∧ {𝑤, 𝑧} = {(𝑓𝑢), 𝑢}) → (𝑤 = (𝑓𝑢) ∧ 𝑧 = 𝑢))
7371, 72syl6bir 257 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑣 = {(𝑓𝑢), 𝑢} → (((𝑤𝑧 ∧ (𝑓𝑢) ∈ 𝑢) ∧ (𝑧𝑣𝑤𝑣)) → (𝑤 = (𝑓𝑢) ∧ 𝑧 = 𝑢)))
7451eqeq2d 2833 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑢 → (𝑤 = (𝑓𝑧) ↔ 𝑤 = (𝑓𝑢)))
7574biimparc 483 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑤 = (𝑓𝑢) ∧ 𝑧 = 𝑢) → 𝑤 = (𝑓𝑧))
7673, 75syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑣 = {(𝑓𝑢), 𝑢} → (((𝑤𝑧 ∧ (𝑓𝑢) ∈ 𝑢) ∧ (𝑧𝑣𝑤𝑣)) → 𝑤 = (𝑓𝑧)))
7776exp4c 436 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = {(𝑓𝑢), 𝑢} → (𝑤𝑧 → ((𝑓𝑢) ∈ 𝑢 → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))))
7877com13 88 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓𝑢) ∈ 𝑢 → (𝑤𝑧 → (𝑣 = {(𝑓𝑢), 𝑢} → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))))
7956, 78syl8 76 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑢𝑥 → (𝑢 ≠ ∅ → (𝑤𝑧 → (𝑣 = {(𝑓𝑢), 𝑢} → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))))))
8079com4r 94 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤𝑧 → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑢𝑥 → (𝑢 ≠ ∅ → (𝑣 = {(𝑓𝑢), 𝑢} → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))))))
8180imp 410 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤𝑧 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑢𝑥 → (𝑢 ≠ ∅ → (𝑣 = {(𝑓𝑢), 𝑢} → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧))))))
8281imp4a 426 . . . . . . . . . . . . . . . . . . . 20 ((𝑤𝑧 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑢𝑥 → ((𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓𝑢), 𝑢}) → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))))
8382com3l 89 . . . . . . . . . . . . . . . . . . 19 (𝑢𝑥 → ((𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓𝑢), 𝑢}) → ((𝑤𝑧 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))))
8483rexlimiv 3266 . . . . . . . . . . . . . . . . . 18 (∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓𝑢), 𝑢}) → ((𝑤𝑧 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧))))
8549, 84sylbi 220 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} → ((𝑤𝑧 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧))))
8685expd 419 . . . . . . . . . . . . . . . 16 (𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} → (𝑤𝑧 → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))))
8786com13 88 . . . . . . . . . . . . . . 15 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑤𝑧 → (𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} → ((𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))))
8887imp4b 425 . . . . . . . . . . . . . 14 ((∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤𝑧) → ((𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} ∧ (𝑧𝑣𝑤𝑣)) → 𝑤 = (𝑓𝑧)))
8988exlimdv 1934 . . . . . . . . . . . . 13 ((∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤𝑧) → (∃𝑣(𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} ∧ (𝑧𝑣𝑤𝑣)) → 𝑤 = (𝑓𝑧)))
9044, 89syl5bi 245 . . . . . . . . . . . 12 ((∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤𝑧) → (∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣) → 𝑤 = (𝑓𝑧)))
9190expimpd 457 . . . . . . . . . . 11 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)) → 𝑤 = (𝑓𝑧)))
9291alrimiv 1928 . . . . . . . . . 10 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∀𝑤((𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)) → 𝑤 = (𝑓𝑧)))
93 mo2icl 3680 . . . . . . . . . 10 (∀𝑤((𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)) → 𝑤 = (𝑓𝑧)) → ∃*𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
9492, 93syl 17 . . . . . . . . 9 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃*𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
9543, 94jctird 530 . . . . . . . 8 (∀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)))))
96 df-reu 3137 . . . . . . . . 9 (∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣) ↔ ∃!𝑤(𝑤𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
97 df-eu 2653 . . . . . . . . 9 (∃!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))))
9896, 97bitri 278 . . . . . . . 8 (∃!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))))
9995, 98syl6ibr 255 . . . . . . 7 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝑧𝑥𝑧 ≠ ∅) → ∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
10099expd 419 . . . . . 6 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑧𝑥 → (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣))))
1012, 100ralrimi 3205 . . . . 5 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
102 vex 3472 . . . . . . . . . . 11 𝑓 ∈ V
103102rnex 7603 . . . . . . . . . 10 ran 𝑓 ∈ V
104 p0ex 5262 . . . . . . . . . 10 {∅} ∈ V
105103, 104unex 7454 . . . . . . . . 9 (ran 𝑓 ∪ {∅}) ∈ V
106 vex 3472 . . . . . . . . 9 𝑥 ∈ V
107105, 106unex 7454 . . . . . . . 8 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∈ V
108107pwex 5258 . . . . . . 7 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∈ V
109 ssun1 4123 . . . . . . . . . . . . . 14 (ran 𝑓 ∪ {∅}) ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
110 fvrn0 6680 . . . . . . . . . . . . . 14 (𝑓𝑢) ∈ (ran 𝑓 ∪ {∅})
111109, 110sselii 3939 . . . . . . . . . . . . 13 (𝑓𝑢) ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
112 elun2 4128 . . . . . . . . . . . . 13 (𝑢𝑥𝑢 ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
113 prssi 4727 . . . . . . . . . . . . 13 (((𝑓𝑢) ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∧ 𝑢 ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)) → {(𝑓𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
114111, 112, 113sylancr 590 . . . . . . . . . . . 12 (𝑢𝑥 → {(𝑓𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
115 prex 5310 . . . . . . . . . . . . 13 {(𝑓𝑢), 𝑢} ∈ V
116115elpw 4515 . . . . . . . . . . . 12 ({(𝑓𝑢), 𝑢} ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ↔ {(𝑓𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
117114, 116sylibr 237 . . . . . . . . . . 11 (𝑢𝑥 → {(𝑓𝑢), 𝑢} ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
118 eleq1 2901 . . . . . . . . . . 11 (𝑔 = {(𝑓𝑢), 𝑢} → (𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ↔ {(𝑓𝑢), 𝑢} ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)))
119117, 118syl5ibrcom 250 . . . . . . . . . 10 (𝑢𝑥 → (𝑔 = {(𝑓𝑢), 𝑢} → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)))
120119adantld 494 . . . . . . . . 9 (𝑢𝑥 → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢}) → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)))
121120rexlimiv 3266 . . . . . . . 8 (∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢}) → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
122121abssi 4021 . . . . . . 7 {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} ⊆ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
123108, 122ssexi 5202 . . . . . 6 {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} ∈ V
124 rexeq 3387 . . . . . . . . 9 (𝑦 = {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} → (∃𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
125124reubidv 3370 . . . . . . . 8 (𝑦 = {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} → (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)))
126125imbi2d 344 . . . . . . 7 (𝑦 = {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} → ((𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) ↔ (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣))))
127126ralbidv 3187 . . . . . 6 (𝑦 = {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} → (∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣))))
128123, 127spcev 3582 . . . . 5 (∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣 ∈ {𝑔 ∣ ∃𝑢𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓𝑢), 𝑢})} (𝑧𝑣𝑤𝑣)) → ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
129101, 128syl 17 . . . 4 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
130129exlimiv 1931 . . 3 (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
131130alimi 1813 . 2 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
1321, 131sylbi 220 1 (CHOICE → ∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2114  ∃*wmo 2620  ∃!weu 2652  {cab 2800   ≠ wne 3011  ∀wral 3130  ∃wrex 3131  ∃!wreu 3132  Vcvv 3469   ∪ cun 3906   ⊆ wss 3908  ∅c0 4265  𝒫 cpw 4511  {csn 4539  {cpr 4541  ran crn 5533  ‘cfv 6334  CHOICEwac 9530 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-reg 9044 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-eprel 5442  df-fr 5491  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-fv 6342  df-ac 9531 This theorem is referenced by:  dfac2  9546
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