Theorem dfac2b 10130

Description: Axiom of Choice (first form) of [Enderton] p. 49 implies our Axiom of Choice (in the form of ac3 10461). The proof does not make use of AC. Note that the Axiom of Regularity is used by the proof. Specifically, elneq 9570 and preleq 9592 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 10129.) (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 10121	. 2 ⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
2		nfra1 3291	. . . . . 6 ⊢ Ⅎ𝑧∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
3		rsp 3255	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑧 ∈ 𝑥 → (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
4		equid 2045	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 = 𝑧
5		neeq1 3022	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → (𝑢 ≠ ∅ ↔ 𝑧 ≠ ∅))
6		eqeq1 2769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → (𝑢 = 𝑧 ↔ 𝑧 = 𝑧))
7	5, 6	anbi12d 644	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑧 → ((𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) ↔ (𝑧 ≠ ∅ ∧ 𝑧 = 𝑧)))
8	7	rspcev 3583	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝑥 ∧ (𝑧 ≠ ∅ ∧ 𝑧 = 𝑧)) → ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧))
9	4, 8	mpanr2 717	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧))
10		fveq2 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧))
11	10	preq1d 4707	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → {(𝑓‘𝑢), 𝑢} = {(𝑓‘𝑧), 𝑢})
12		preq2 4702	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → {(𝑓‘𝑧), 𝑢} = {(𝑓‘𝑧), 𝑧})
13	11, 12	eqtr2d 2801	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑧 → {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢})
14	13	anim2i 629	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) → (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢}))
15	14	reximi 3105	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) → ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢}))
16	9, 15	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢}))
17		prex 5411	. . . . . . . . . . . . . . . . 17 ⊢ {(𝑓‘𝑧), 𝑧} ∈ V
18		eqeq1 2769	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = {(𝑓‘𝑧), 𝑧} → (𝑔 = {(𝑓‘𝑢), 𝑢} ↔ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢}))
19	18	anbi2d 642	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = {(𝑓‘𝑧), 𝑧} → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢})))
20	19	rexbidv 3191	. . . . . . . . . . . . . . . . 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 3640	. . . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → {(𝑓‘𝑧), 𝑧} ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})})
23		vex 3461	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
24	23	prid2 4731	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ {(𝑓‘𝑧), 𝑧}
25		fvex 6898	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓‘𝑧) ∈ V
26	25	prid1 4730	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧}
27	24, 26	pm3.2i 476	. . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = {(𝑓‘𝑧), 𝑧} → ((𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣) ↔ (𝑧 ∈ {(𝑓‘𝑧), 𝑧} ∧ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧})))
31	30	rspcev 3583	. . . . . . . . . . . . . . 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 598	. . . . . . . . . . . . . 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 642	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑓‘𝑧) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣)))
36	35	rexbidv 3191	. . . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . . 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 3567	. . . . . . . . . . . . . 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 605	. . . . . . . . . . . . 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 418	. . . . . . . . . . . 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 77	. . . . . . . . . . 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 416	. . . . . . . . . 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 54	. . . . . . . . 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 3092	. . . . . . . . . . . . 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 3461	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑣 ∈ V
46		eqeq1 2769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = 𝑣 → (𝑔 = {(𝑓‘𝑢), 𝑢} ↔ 𝑣 = {(𝑓‘𝑢), 𝑢}))
47	46	anbi2d 642	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = 𝑣 → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢})))
48	47	rexbidv 3191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑣 → (∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢})))
49	45, 48	elab 3640	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ↔ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢}))
50		neeq1 3022	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = 𝑢 → (𝑧 ≠ ∅ ↔ 𝑢 ≠ ∅))
51		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 282	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = 𝑢 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑢) ∈ 𝑢))
55	50, 54	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 = 𝑢 → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑢 ≠ ∅ → (𝑓‘𝑢) ∈ 𝑢)))
56	55	rspccv 3580	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑢 ∈ 𝑥 → (𝑢 ≠ ∅ → (𝑓‘𝑢) ∈ 𝑢)))
57		elneq 9570	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 ∈ 𝑧 → 𝑤 ≠ 𝑧)
58	57	neneqd 2965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 ∈ 𝑧 → ¬ 𝑤 = 𝑧)
59		vex 3461	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ 𝑤 ∈ V
60		neqne 2968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑤 = 𝑧 → 𝑤 ≠ 𝑧)
61		prel12g 4831	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (¬ 𝑤 = 𝑧 → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑤 ∈ {(𝑓‘𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓‘𝑢), 𝑢})))
63		eleq2 2854	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ {(𝑓‘𝑢), 𝑢}))
64		eleq2 2854	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (𝑧 ∈ 𝑣 ↔ 𝑧 ∈ {(𝑓‘𝑢), 𝑢}))
65	63, 64	anbi12d 644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑤 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣) ↔ (𝑤 ∈ {(𝑓‘𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓‘𝑢), 𝑢})))
66		ancom 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑤 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
67	65, 66	bitr3di 289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑤 ∈ {(𝑓‘𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓‘𝑢), 𝑢}) ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
68	62, 67	sylan9bbr 520	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑣 = {(𝑓‘𝑢), 𝑢} ∧ ¬ 𝑤 = 𝑧) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
69	58, 68	sylan2 605	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑣 = {(𝑓‘𝑢), 𝑢} ∧ 𝑤 ∈ 𝑧) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
70	69	adantrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑣 = {(𝑓‘𝑢), 𝑢} ∧ (𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢)) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
71	70	pm5.32da 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9592	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢) ∧ {𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢}) → (𝑤 = (𝑓‘𝑢) ∧ 𝑧 = 𝑢))
73	71, 72	biimtrrdi 257	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (((𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢) ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → (𝑤 = (𝑓‘𝑢) ∧ 𝑧 = 𝑢)))
74	51	eqeq2d 2776	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = 𝑢 → (𝑤 = (𝑓‘𝑧) ↔ 𝑤 = (𝑓‘𝑢)))
75	74	biimparc 485	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑤 = (𝑓‘𝑢) ∧ 𝑧 = 𝑢) → 𝑤 = (𝑓‘𝑧))
76	73, 75	syl6 36	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (((𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢) ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑤 = (𝑓‘𝑧)))
77	76	exp4c 438	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (𝑤 ∈ 𝑧 → ((𝑓‘𝑢) ∈ 𝑢 → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧)))))
78	77	com13 89	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓‘𝑢) ∈ 𝑢 → (𝑤 ∈ 𝑧 → (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑤 = (𝑓‘𝑧)))))
79	56, 78	syl8 77	. . . . . . . . . . . . . . . . . . . . . . 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 95	. . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . 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 428	. . . . . . . . . . . . . . . . . . . 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 90	. . . . . . . . . . . . . . . . . . 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 3161	. . . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . . 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 421	. . . . . . . . . . . . . . . 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 89	. . . . . . . . . . . . . . 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 427	. . . . . . . . . . . . . 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 1966	. . . . . . . . . . . . 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	biimtrid 245	. . . . . . . . . . . 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 459	. . . . . . . . . . 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 1960	. . . . . . . . . 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 3679	. . . . . . . . . 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 18	. . . . . . . . 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 536	. . . . . . . 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 3372	. . . . . . . . 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 2599	. . . . . . . . 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 278	. . . . . . . 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	imbitrrdi 255	. . . . . . 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 421	. . . . . 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 3265	. . . . 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 3461	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
103	102	rnex 7913	. . . . . . . . . 10 ⊢ ran 𝑓 ∈ V
104		p0ex 5357	. . . . . . . . . 10 ⊢ {∅} ∈ V
105	103, 104	unex 7752	. . . . . . . . 9 ⊢ (ran 𝑓 ∪ {∅}) ∈ V
106		vex 3461	. . . . . . . . 9 ⊢ 𝑥 ∈ V
107	105, 106	unex 7752	. . . . . . . 8 ⊢ ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∈ V
108	107	pwex 5353	. . . . . . 7 ⊢ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∈ V
109		ssun1 4131	. . . . . . . . . . . . . 14 ⊢ (ran 𝑓 ∪ {∅}) ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
110		fvrn0 6913	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑢) ∈ (ran 𝑓 ∪ {∅})
111	109, 110	sselii 3935	. . . . . . . . . . . . 13 ⊢ (𝑓‘𝑢) ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
112		elun2 4136	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ 𝑥 → 𝑢 ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
113		prssi 4789	. . . . . . . . . . . . 13 ⊢ (((𝑓‘𝑢) ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∧ 𝑢 ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)) → {(𝑓‘𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
114	111, 112, 113	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝑥 → {(𝑓‘𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
115		prex 5411	. . . . . . . . . . . . 13 ⊢ {(𝑓‘𝑢), 𝑢} ∈ V
116	115	elpw 4568	. . . . . . . . . . . 12 ⊢ ({(𝑓‘𝑢), 𝑢} ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ↔ {(𝑓‘𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
117	114, 116	sylibr 237	. . . . . . . . . . 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 250	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑥 → (𝑔 = {(𝑓‘𝑢), 𝑢} → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)))
120	119	adantld 496	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝑥 → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)))
121	120	rexlimiv 3161	. . . . . . . 8 ⊢ (∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
122	121	abssi 4023	. . . . . . 7 ⊢ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ⊆ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
123	108, 122	ssexi 5295	. . . . . 6 ⊢ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ∈ V
124		rexeq 3321	. . . . . . . . 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 3387	. . . . . . . 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 343	. . . . . . 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 3190	. . . . . 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 3567	. . . . 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 18	. . . 4 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
130	129	exlimiv 1963	. . 3 ⊢ (∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
131	130	alimi 1844	. 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 220	1 ⊢ (CHOICE → ∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∃*wmo 2567  ∃!weu 2598  {cab 2743   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  ∃!wreu 3369  Vcvv 3457   ∪ cun 3904   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564  {csn 4591  {cpr 4593  ran crn 5664  ‘cfv 6540  CHOICEwac 10115

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-reg 9561

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-eprel 5563  df-fr 5616  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548  df-ac 10116

This theorem is used by:  dfac2  10131