Theorem dfac2b 9352

Description: Axiom of Choice (first form) of [Enderton] p. 49 implies our Axiom of Choice (in the form of ac3 9684). The proof does not make use of AC. Note that the Axiom of Regularity is used by the proof. Specifically, elneq 8859 and preleq 8875 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 9351.) (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by AV, 16-Jun-2022.)

Assertion

Ref	Expression
dfac2b	⊢ (CHOICE → ∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))

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.

Step	Hyp	Ref	Expression
1		dfac3 9343	. 2 ⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
2		nfra1 3169	. . . . . 6 ⊢ Ⅎ𝑧∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
3		rsp 3155	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑧 ∈ 𝑥 → (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
4		equid 1969	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 = 𝑧
5		neeq1 3029	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → (𝑢 ≠ ∅ ↔ 𝑧 ≠ ∅))
6		eqeq1 2782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → (𝑢 = 𝑧 ↔ 𝑧 = 𝑧))
7	5, 6	anbi12d 621	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑧 → ((𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) ↔ (𝑧 ≠ ∅ ∧ 𝑧 = 𝑧)))
8	7	rspcev 3535	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝑥 ∧ (𝑧 ≠ ∅ ∧ 𝑧 = 𝑧)) → ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧))
9	4, 8	mpanr2 691	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧))
10		fveq2 6501	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧))
11	10	preq1d 4550	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → {(𝑓‘𝑢), 𝑢} = {(𝑓‘𝑧), 𝑢})
12		preq2 4545	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → {(𝑓‘𝑧), 𝑢} = {(𝑓‘𝑧), 𝑧})
13	11, 12	eqtr2d 2815	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑧 → {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢})
14	13	anim2i 607	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) → (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢}))
15	14	reximi 3190	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) → ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢}))
16	9, 15	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢}))
17		prex 5190	. . . . . . . . . . . . . . . . 17 ⊢ {(𝑓‘𝑧), 𝑧} ∈ V
18		eqeq1 2782	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = {(𝑓‘𝑧), 𝑧} → (𝑔 = {(𝑓‘𝑢), 𝑢} ↔ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢}))
19	18	anbi2d 619	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = {(𝑓‘𝑧), 𝑧} → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢})))
20	19	rexbidv 3242	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = {(𝑓‘𝑧), 𝑧} → (∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢})))
21	17, 20	elab 3582	. . . . . . . . . . . . . . . 16 ⊢ ({(𝑓‘𝑧), 𝑧} ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ↔ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢}))
22	16, 21	sylibr 226	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → {(𝑓‘𝑧), 𝑧} ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})})
23		vex 3418	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
24	23	prid2 4574	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ {(𝑓‘𝑧), 𝑧}
25		fvex 6514	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓‘𝑧) ∈ V
26	25	prid1 4573	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧}
27	24, 26	pm3.2i 463	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {(𝑓‘𝑧), 𝑧} ∧ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧})
28		eleq2 2854	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = {(𝑓‘𝑧), 𝑧} → (𝑧 ∈ 𝑣 ↔ 𝑧 ∈ {(𝑓‘𝑧), 𝑧}))
29		eleq2 2854	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = {(𝑓‘𝑧), 𝑧} → ((𝑓‘𝑧) ∈ 𝑣 ↔ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧}))
30	28, 29	anbi12d 621	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = {(𝑓‘𝑧), 𝑧} → ((𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣) ↔ (𝑧 ∈ {(𝑓‘𝑧), 𝑧} ∧ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧})))
31	30	rspcev 3535	. . . . . . . . . . . . . . 15 ⊢ (({(𝑓‘𝑧), 𝑧} ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ∧ (𝑧 ∈ {(𝑓‘𝑧), 𝑧} ∧ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧})) → ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣))
32	22, 27, 31	sylancl 577	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣))
33		eleq1 2853	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑓‘𝑧) → (𝑤 ∈ 𝑧 ↔ (𝑓‘𝑧) ∈ 𝑧))
34		eleq1 2853	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑓‘𝑧) → (𝑤 ∈ 𝑣 ↔ (𝑓‘𝑧) ∈ 𝑣))
35	34	anbi2d 619	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑓‘𝑧) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣)))
36	35	rexbidv 3242	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑓‘𝑧) → (∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣)))
37	33, 36	anbi12d 621	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑓‘𝑧) → ((𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ↔ ((𝑓‘𝑧) ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣))))
38	25, 37	spcev 3525	. . . . . . . . . . . . . 14 ⊢ (((𝑓‘𝑧) ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣)) → ∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
39	32, 38	sylan2 583	. . . . . . . . . . . . 13 ⊢ (((𝑓‘𝑧) ∈ 𝑧 ∧ (𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅)) → ∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
40	39	ex 405	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑧) ∈ 𝑧 → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))))
41	3, 40	syl8 76	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑧 ∈ 𝑥 → (𝑧 ≠ ∅ → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))))))
42	41	impd 402	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))))
43	42	pm2.43d 53	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))))
44		df-rex 3094	. . . . . . . . . . . . 13 ⊢ (∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑣(𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
45		vex 3418	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑣 ∈ V
46		eqeq1 2782	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = 𝑣 → (𝑔 = {(𝑓‘𝑢), 𝑢} ↔ 𝑣 = {(𝑓‘𝑢), 𝑢}))
47	46	anbi2d 619	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = 𝑣 → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢})))
48	47	rexbidv 3242	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑣 → (∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢})))
49	45, 48	elab 3582	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ↔ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢}))
50		neeq1 3029	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = 𝑢 → (𝑧 ≠ ∅ ↔ 𝑢 ≠ ∅))
51		fveq2 6501	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = 𝑢 → (𝑓‘𝑧) = (𝑓‘𝑢))
52	51	eleq1d 2850	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 = 𝑢 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑢) ∈ 𝑧))
53		eleq2 2854	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 = 𝑢 → ((𝑓‘𝑢) ∈ 𝑧 ↔ (𝑓‘𝑢) ∈ 𝑢))
54	52, 53	bitrd 271	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = 𝑢 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑢) ∈ 𝑢))
55	50, 54	imbi12d 337	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 = 𝑢 → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑢 ≠ ∅ → (𝑓‘𝑢) ∈ 𝑢)))
56	55	rspccv 3532	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑢 ∈ 𝑥 → (𝑢 ≠ ∅ → (𝑓‘𝑢) ∈ 𝑢)))
57		elneq 8859	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 ∈ 𝑧 → 𝑤 ≠ 𝑧)
58	57	neneqd 2972	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 ∈ 𝑧 → ¬ 𝑤 = 𝑧)
59		vex 3418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ 𝑤 ∈ V
60		neqne 2975	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑤 = 𝑧 → 𝑤 ≠ 𝑧)
61		prel12g 4669	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑤 ∈ V ∧ 𝑧 ∈ V ∧ 𝑤 ≠ 𝑧) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑤 ∈ {(𝑓‘𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓‘𝑢), 𝑢})))
62	59, 23, 60, 61	mp3an12i 1444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (¬ 𝑤 = 𝑧 → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑤 ∈ {(𝑓‘𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓‘𝑢), 𝑢})))
63		ancom 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑤 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
64		eleq2 2854	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ {(𝑓‘𝑢), 𝑢}))
65		eleq2 2854	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (𝑧 ∈ 𝑣 ↔ 𝑧 ∈ {(𝑓‘𝑢), 𝑢}))
66	64, 65	anbi12d 621	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑤 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣) ↔ (𝑤 ∈ {(𝑓‘𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓‘𝑢), 𝑢})))
67	63, 66	syl5rbbr 278	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑤 ∈ {(𝑓‘𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓‘𝑢), 𝑢}) ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
68	62, 67	sylan9bbr 503	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑣 = {(𝑓‘𝑢), 𝑢} ∧ ¬ 𝑤 = 𝑧) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
69	58, 68	sylan2 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑣 = {(𝑓‘𝑢), 𝑢} ∧ 𝑤 ∈ 𝑧) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
70	69	adantrr 704	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑣 = {(𝑓‘𝑢), 𝑢} ∧ (𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢)) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
71	70	pm5.32da 571	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (((𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢) ∧ {𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢}) ↔ ((𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢) ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))))
72	23	preleq 8875	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢) ∧ {𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢}) → (𝑤 = (𝑓‘𝑢) ∧ 𝑧 = 𝑢))
73	71, 72	syl6bir 246	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (((𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢) ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → (𝑤 = (𝑓‘𝑢) ∧ 𝑧 = 𝑢)))
74	51	eqeq2d 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = 𝑢 → (𝑤 = (𝑓‘𝑧) ↔ 𝑤 = (𝑓‘𝑢)))
75	74	biimparc 472	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑤 = (𝑓‘𝑢) ∧ 𝑧 = 𝑢) → 𝑤 = (𝑓‘𝑧))
76	73, 75	syl6 35	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (((𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢) ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑤 = (𝑓‘𝑧)))
77	76	exp4c 425	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (𝑤 ∈ 𝑧 → ((𝑓‘𝑢) ∈ 𝑢 → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧)))))
78	77	com13 88	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓‘𝑢) ∈ 𝑢 → (𝑤 ∈ 𝑧 → (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧)))))
79	56, 78	syl8 76	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑢 ∈ 𝑥 → (𝑢 ≠ ∅ → (𝑤 ∈ 𝑧 → (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧)))))))
80	79	com4r 94	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ 𝑧 → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑢 ∈ 𝑥 → (𝑢 ≠ ∅ → (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧)))))))
81	80	imp 398	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ 𝑧 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → (𝑢 ∈ 𝑥 → (𝑢 ≠ ∅ → (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧))))))
82	81	imp4a 415	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ 𝑧 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → (𝑢 ∈ 𝑥 → ((𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢}) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧)))))
83	82	com3l 89	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ 𝑥 → ((𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢}) → ((𝑤 ∈ 𝑧 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧)))))
84	83	rexlimiv 3225	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢}) → ((𝑤 ∈ 𝑧 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧))))
85	49, 84	sylbi 209	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} → ((𝑤 ∈ 𝑧 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧))))
86	85	expd 408	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} → (𝑤 ∈ 𝑧 → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧)))))
87	86	com13 88	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑤 ∈ 𝑧 → (𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧)))))
88	87	imp4b 414	. . . . . . . . . . . . . 14 ⊢ ((∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ 𝑤 ∈ 𝑧) → ((𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑤 = (𝑓‘𝑧)))
89	88	exlimdv 1892	. . . . . . . . . . . . 13 ⊢ ((∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ 𝑤 ∈ 𝑧) → (∃𝑣(𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑤 = (𝑓‘𝑧)))
90	44, 89	syl5bi 234	. . . . . . . . . . . 12 ⊢ ((∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ 𝑤 ∈ 𝑧) → (∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧)))
91	90	expimpd 446	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ((𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑤 = (𝑓‘𝑧)))
92	91	alrimiv 1886	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∀𝑤((𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑤 = (𝑓‘𝑧)))
93		mo2icl 3621	. . . . . . . . . 10 ⊢ (∀𝑤((𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑤 = (𝑓‘𝑧)) → ∃*𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
94	92, 93	syl 17	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃*𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
95	43, 94	jctird 519	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → (∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ∧ ∃*𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))))
96		df-reu 3095	. . . . . . . . 9 ⊢ (∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃!𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
97		df-eu 2583	. . . . . . . . 9 ⊢ (∃!𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ↔ (∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ∧ ∃*𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))))
98	96, 97	bitri 267	. . . . . . . 8 ⊢ (∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (∃𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ∧ ∃*𝑤(𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))))
99	95, 98	syl6ibr 244	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
100	99	expd 408	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑧 ∈ 𝑥 → (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))))
101	2, 100	ralrimi 3166	. . . . 5 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
102		vex 3418	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
103	102	rnex 7434	. . . . . . . . . 10 ⊢ ran 𝑓 ∈ V
104		p0ex 5138	. . . . . . . . . 10 ⊢ {∅} ∈ V
105	103, 104	unex 7288	. . . . . . . . 9 ⊢ (ran 𝑓 ∪ {∅}) ∈ V
106		vex 3418	. . . . . . . . 9 ⊢ 𝑥 ∈ V
107	105, 106	unex 7288	. . . . . . . 8 ⊢ ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∈ V
108	107	pwex 5135	. . . . . . 7 ⊢ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∈ V
109		ssun1 4039	. . . . . . . . . . . . . 14 ⊢ (ran 𝑓 ∪ {∅}) ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
110		fvrn0 6529	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑢) ∈ (ran 𝑓 ∪ {∅})
111	109, 110	sselii 3857	. . . . . . . . . . . . 13 ⊢ (𝑓‘𝑢) ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
112		elun2 4044	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ 𝑥 → 𝑢 ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
113		prssi 4629	. . . . . . . . . . . . 13 ⊢ (((𝑓‘𝑢) ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∧ 𝑢 ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)) → {(𝑓‘𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
114	111, 112, 113	sylancr 578	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝑥 → {(𝑓‘𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
115		prex 5190	. . . . . . . . . . . . 13 ⊢ {(𝑓‘𝑢), 𝑢} ∈ V
116	115	elpw 4429	. . . . . . . . . . . 12 ⊢ ({(𝑓‘𝑢), 𝑢} ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ↔ {(𝑓‘𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
117	114, 116	sylibr 226	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝑥 → {(𝑓‘𝑢), 𝑢} ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
118		eleq1 2853	. . . . . . . . . . 11 ⊢ (𝑔 = {(𝑓‘𝑢), 𝑢} → (𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ↔ {(𝑓‘𝑢), 𝑢} ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)))
119	117, 118	syl5ibrcom 239	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑥 → (𝑔 = {(𝑓‘𝑢), 𝑢} → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)))
120	119	adantld 483	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝑥 → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)))
121	120	rexlimiv 3225	. . . . . . . 8 ⊢ (∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
122	121	abssi 3938	. . . . . . 7 ⊢ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ⊆ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
123	108, 122	ssexi 5083	. . . . . 6 ⊢ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ∈ V
124		rexeq 3346	. . . . . . . . 9 ⊢ (𝑦 = {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} → (∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
125	124	reubidv 3329	. . . . . . . 8 ⊢ (𝑦 = {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} → (∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
126	125	imbi2d 333	. . . . . . 7 ⊢ (𝑦 = {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} → ((𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ↔ (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))))
127	126	ralbidv 3147	. . . . . 6 ⊢ (𝑦 = {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ↔ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))))
128	123, 127	spcev 3525	. . . . 5 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
129	101, 128	syl 17	. . . 4 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
130	129	exlimiv 1889	. . 3 ⊢ (∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
131	130	alimi 1774	. 2 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
132	1, 131	sylbi 209	1 ⊢ (CHOICE → ∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387  ∀wal 1505   = wceq 1507  ∃wex 1742   ∈ wcel 2050  ∃*wmo 2545  ∃!weu 2582  {cab 2758   ≠ wne 2967  ∀wral 3088  ∃wrex 3089  ∃!wreu 3090  Vcvv 3415   ∪ cun 3829   ⊆ wss 3831  ∅c0 4180  𝒫 cpw 4423  {csn 4442  {cpr 4444  ran crn 5409  ‘cfv 6190  CHOICEwac 9337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-reg 8853

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-eprel 5318  df-fr 5367  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fn 6193  df-fv 6198  df-ac 9338

This theorem is referenced by:  dfac2  9353