Theorem dfac2b 10044

Description: Axiom of Choice (first form) of [Enderton] p. 49 implies our Axiom of Choice (in the form of ac3 10375). The proof does not make use of AC. Note that the Axiom of Regularity is used by the proof. Specifically, elneq 9506 and preleq 9528 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 10043.) (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 10034	. 2 ⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
2		nfra1 3263	. . . . . 6 ⊢ Ⅎ𝑧∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
3		rsp 3227	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑧 ∈ 𝑥 → (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
4		equid 2019	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 = 𝑧
5		neeq1 2996	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → (𝑢 ≠ ∅ ↔ 𝑧 ≠ ∅))
6		eqeq1 2743	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → (𝑢 = 𝑧 ↔ 𝑧 = 𝑧))
7	5, 6	anbi12d 638	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑧 → ((𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) ↔ (𝑧 ≠ ∅ ∧ 𝑧 = 𝑧)))
8	7	rspcev 3560	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝑥 ∧ (𝑧 ≠ ∅ ∧ 𝑧 = 𝑧)) → ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧))
9	4, 8	mpanr2 710	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧))
10		fveq2 6827	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧))
11	10	preq1d 4671	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → {(𝑓‘𝑢), 𝑢} = {(𝑓‘𝑧), 𝑢})
12		preq2 4666	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → {(𝑓‘𝑧), 𝑢} = {(𝑓‘𝑧), 𝑧})
13	11, 12	eqtr2d 2775	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑧 → {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢})
14	13	anim2i 623	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) → (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢}))
15	14	reximi 3077	. . . . . . . . . . . . . . . . 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 5367	. . . . . . . . . . . . . . . . 17 ⊢ {(𝑓‘𝑧), 𝑧} ∈ V
18		eqeq1 2743	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = {(𝑓‘𝑧), 𝑧} → (𝑔 = {(𝑓‘𝑢), 𝑢} ↔ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢}))
19	18	anbi2d 636	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = {(𝑓‘𝑧), 𝑧} → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢})))
20	19	rexbidv 3163	. . . . . . . . . . . . . . . . 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 3617	. . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → {(𝑓‘𝑧), 𝑧} ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})})
23		vex 3435	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
24	23	prid2 4695	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ {(𝑓‘𝑧), 𝑧}
25		fvex 6840	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓‘𝑧) ∈ V
26	25	prid1 4694	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧}
27	24, 26	pm3.2i 471	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {(𝑓‘𝑧), 𝑧} ∧ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧})
28		eleq2 2828	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = {(𝑓‘𝑧), 𝑧} → (𝑧 ∈ 𝑣 ↔ 𝑧 ∈ {(𝑓‘𝑧), 𝑧}))
29		eleq2 2828	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = {(𝑓‘𝑧), 𝑧} → ((𝑓‘𝑧) ∈ 𝑣 ↔ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧}))
30	28, 29	anbi12d 638	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = {(𝑓‘𝑧), 𝑧} → ((𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣) ↔ (𝑧 ∈ {(𝑓‘𝑧), 𝑧} ∧ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧})))
31	30	rspcev 3560	. . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣))
33		eleq1 2827	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑓‘𝑧) → (𝑤 ∈ 𝑧 ↔ (𝑓‘𝑧) ∈ 𝑧))
34		eleq1 2827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑓‘𝑧) → (𝑤 ∈ 𝑣 ↔ (𝑓‘𝑧) ∈ 𝑣))
35	34	anbi2d 636	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑓‘𝑧) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣)))
36	35	rexbidv 3163	. . . . . . . . . . . . . . . 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 638	. . . . . . . . . . . . . . 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 3544	. . . . . . . . . . . . . 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 599	. . . . . . . . . . . . 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 413	. . . . . . . . . . . 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 411	. . . . . . . . . 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 3064	. . . . . . . . . . . . 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 3435	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑣 ∈ V
46		eqeq1 2743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = 𝑣 → (𝑔 = {(𝑓‘𝑢), 𝑢} ↔ 𝑣 = {(𝑓‘𝑢), 𝑢}))
47	46	anbi2d 636	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = 𝑣 → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢})))
48	47	rexbidv 3163	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑣 → (∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢})))
49	45, 48	elab 3617	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ↔ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢}))
50		neeq1 2996	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = 𝑢 → (𝑧 ≠ ∅ ↔ 𝑢 ≠ ∅))
51		fveq2 6827	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = 𝑢 → (𝑓‘𝑧) = (𝑓‘𝑢))
52	51	eleq1d 2824	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 = 𝑢 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑢) ∈ 𝑧))
53		eleq2 2828	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 = 𝑢 → ((𝑓‘𝑢) ∈ 𝑧 ↔ (𝑓‘𝑢) ∈ 𝑢))
54	52, 53	bitrd 280	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = 𝑢 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑢) ∈ 𝑢))
55	50, 54	imbi12d 345	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 = 𝑢 → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑢 ≠ ∅ → (𝑓‘𝑢) ∈ 𝑢)))
56	55	rspccv 3557	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑢 ∈ 𝑥 → (𝑢 ≠ ∅ → (𝑓‘𝑢) ∈ 𝑢)))
57		elneq 9506	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 ∈ 𝑧 → 𝑤 ≠ 𝑧)
58	57	neneqd 2939	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 ∈ 𝑧 → ¬ 𝑤 = 𝑧)
59		vex 3435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ 𝑤 ∈ V
60		neqne 2942	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑤 = 𝑧 → 𝑤 ≠ 𝑧)
61		prel12g 4795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (¬ 𝑤 = 𝑧 → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑤 ∈ {(𝑓‘𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓‘𝑢), 𝑢})))
63		eleq2 2828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ {(𝑓‘𝑢), 𝑢}))
64		eleq2 2828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (𝑧 ∈ 𝑣 ↔ 𝑧 ∈ {(𝑓‘𝑢), 𝑢}))
65	63, 64	anbi12d 638	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑤 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣) ↔ (𝑤 ∈ {(𝑓‘𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓‘𝑢), 𝑢})))
66		ancom 461	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑤 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
67	65, 66	bitr3di 287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑤 ∈ {(𝑓‘𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓‘𝑢), 𝑢}) ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
68	62, 67	sylan9bbr 515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑣 = {(𝑓‘𝑢), 𝑢} ∧ ¬ 𝑤 = 𝑧) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
69	58, 68	sylan2 599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑣 = {(𝑓‘𝑢), 𝑢} ∧ 𝑤 ∈ 𝑧) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
70	69	adantrr 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑣 = {(𝑓‘𝑢), 𝑢} ∧ (𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢)) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
71	70	pm5.32da 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9528	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢) ∧ {𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢}) → (𝑤 = (𝑓‘𝑢) ∧ 𝑧 = 𝑢))
73	71, 72	biimtrrdi 255	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (((𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢) ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → (𝑤 = (𝑓‘𝑢) ∧ 𝑧 = 𝑢)))
74	51	eqeq2d 2750	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = 𝑢 → (𝑤 = (𝑓‘𝑧) ↔ 𝑤 = (𝑓‘𝑢)))
75	74	biimparc 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 433	. . . . . . . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . . . . 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 423	. . . . . . . . . . . . . . . . . . . 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 3133	. . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . 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 422	. . . . . . . . . . . . . 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 1940	. . . . . . . . . . . . 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 243	. . . . . . . . . . . 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 454	. . . . . . . . . . 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 1934	. . . . . . . . . 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 3655	. . . . . . . . . 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 531	. . . . . . . 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 3345	. . . . . . . . 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 2573	. . . . . . . . 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 276	. . . . . . . 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 253	. . . . . . 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 416	. . . . . 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 3237	. . . . 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 3435	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
103	102	rnex 7850	. . . . . . . . . 10 ⊢ ran 𝑓 ∈ V
104		p0ex 5313	. . . . . . . . . 10 ⊢ {∅} ∈ V
105	103, 104	unex 7687	. . . . . . . . 9 ⊢ (ran 𝑓 ∪ {∅}) ∈ V
106		vex 3435	. . . . . . . . 9 ⊢ 𝑥 ∈ V
107	105, 106	unex 7687	. . . . . . . 8 ⊢ ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∈ V
108	107	pwex 5309	. . . . . . 7 ⊢ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∈ V
109		ssun1 4107	. . . . . . . . . . . . . 14 ⊢ (ran 𝑓 ∪ {∅}) ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
110		fvrn0 6855	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑢) ∈ (ran 𝑓 ∪ {∅})
111	109, 110	sselii 3912	. . . . . . . . . . . . 13 ⊢ (𝑓‘𝑢) ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
112		elun2 4112	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ 𝑥 → 𝑢 ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
113		prssi 4752	. . . . . . . . . . . . 13 ⊢ (((𝑓‘𝑢) ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∧ 𝑢 ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)) → {(𝑓‘𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
114	111, 112, 113	sylancr 593	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝑥 → {(𝑓‘𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
115		prex 5367	. . . . . . . . . . . . 13 ⊢ {(𝑓‘𝑢), 𝑢} ∈ V
116	115	elpw 4533	. . . . . . . . . . . 12 ⊢ ({(𝑓‘𝑢), 𝑢} ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ↔ {(𝑓‘𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
117	114, 116	sylibr 235	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝑥 → {(𝑓‘𝑢), 𝑢} ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
118		eleq1 2827	. . . . . . . . . . 11 ⊢ (𝑔 = {(𝑓‘𝑢), 𝑢} → (𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ↔ {(𝑓‘𝑢), 𝑢} ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)))
119	117, 118	syl5ibrcom 248	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑥 → (𝑔 = {(𝑓‘𝑢), 𝑢} → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)))
120	119	adantld 491	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝑥 → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)))
121	120	rexlimiv 3133	. . . . . . . 8 ⊢ (∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
122	121	abssi 3999	. . . . . . 7 ⊢ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ⊆ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
123	108, 122	ssexi 5250	. . . . . 6 ⊢ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ∈ V
124		rexeq 3293	. . . . . . . . 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 3360	. . . . . . . 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 341	. . . . . . 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 3162	. . . . . 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 3544	. . . . 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 1937	. . 3 ⊢ (∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
131	130	alimi 1818	. 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 218	1 ⊢ (CHOICE → ∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∃*wmo 2541  ∃!weu 2572  {cab 2717   ≠ wne 2934  ∀wral 3053  ∃wrex 3063  ∃!wreu 3342  Vcvv 3431   ∪ cun 3881   ⊆ wss 3883  ∅c0 4261  𝒫 cpw 4529  {csn 4555  {cpr 4557  ran crn 5619  ‘cfv 6485  CHOICEwac 10028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-reg 9497

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-eprel 5518  df-fr 5571  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-fv 6493  df-ac 10029

This theorem is referenced by:  dfac2  10045