Theorem dfac2b 10110

Description: Axiom of Choice (first form) of [Enderton] p. 49 implies our Axiom of Choice (in the form of ac3 10441). The proof does not make use of AC. Note that the Axiom of Regularity is used by the proof. Specifically, elneq 9559 and preleq 9581 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 10109.) (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 10101	. 2 ⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
2		nfra1 3289	. . . . . 6 ⊢ Ⅎ𝑧∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
3		rsp 3253	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑧 ∈ 𝑥 → (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
4		equid 2042	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 = 𝑧
5		neeq1 3020	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → (𝑢 ≠ ∅ ↔ 𝑧 ≠ ∅))
6		eqeq1 2767	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → (𝑢 = 𝑧 ↔ 𝑧 = 𝑧))
7	5, 6	anbi12d 643	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑧 → ((𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) ↔ (𝑧 ≠ ∅ ∧ 𝑧 = 𝑧)))
8	7	rspcev 3581	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝑥 ∧ (𝑧 ≠ ∅ ∧ 𝑧 = 𝑧)) → ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧))
9	4, 8	mpanr2 716	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑢 = 𝑧))
10		fveq2 6881	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧))
11	10	preq1d 4705	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → {(𝑓‘𝑢), 𝑢} = {(𝑓‘𝑧), 𝑢})
12		preq2 4700	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → {(𝑓‘𝑧), 𝑢} = {(𝑓‘𝑧), 𝑧})
13	11, 12	eqtr2d 2799	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑧 → {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢})
14	13	anim2i 628	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ≠ ∅ ∧ 𝑢 = 𝑧) → (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢}))
15	14	reximi 3103	. . . . . . . . . . . . . . . . 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 5409	. . . . . . . . . . . . . . . . 17 ⊢ {(𝑓‘𝑧), 𝑧} ∈ V
18		eqeq1 2767	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = {(𝑓‘𝑧), 𝑧} → (𝑔 = {(𝑓‘𝑢), 𝑢} ↔ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢}))
19	18	anbi2d 641	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = {(𝑓‘𝑧), 𝑧} → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ (𝑢 ≠ ∅ ∧ {(𝑓‘𝑧), 𝑧} = {(𝑓‘𝑢), 𝑢})))
20	19	rexbidv 3189	. . . . . . . . . . . . . . . . 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 3638	. . . . . . . . . . . . . . . 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 3459	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
24	23	prid2 4729	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ {(𝑓‘𝑧), 𝑧}
25		fvex 6894	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓‘𝑧) ∈ V
26	25	prid1 4728	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧}
27	24, 26	pm3.2i 475	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {(𝑓‘𝑧), 𝑧} ∧ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧})
28		eleq2 2852	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = {(𝑓‘𝑧), 𝑧} → (𝑧 ∈ 𝑣 ↔ 𝑧 ∈ {(𝑓‘𝑧), 𝑧}))
29		eleq2 2852	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = {(𝑓‘𝑧), 𝑧} → ((𝑓‘𝑧) ∈ 𝑣 ↔ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧}))
30	28, 29	anbi12d 643	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = {(𝑓‘𝑧), 𝑧} → ((𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣) ↔ (𝑧 ∈ {(𝑓‘𝑧), 𝑧} ∧ (𝑓‘𝑧) ∈ {(𝑓‘𝑧), 𝑧})))
31	30	rspcev 3581	. . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → ∃𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣))
33		eleq1 2851	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑓‘𝑧) → (𝑤 ∈ 𝑧 ↔ (𝑓‘𝑧) ∈ 𝑧))
34		eleq1 2851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑓‘𝑧) → (𝑤 ∈ 𝑣 ↔ (𝑓‘𝑧) ∈ 𝑣))
35	34	anbi2d 641	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑓‘𝑧) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ (𝑓‘𝑧) ∈ 𝑣)))
36	35	rexbidv 3189	. . . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . . 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 3565	. . . . . . . . . . . . . 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 604	. . . . . . . . . . . . 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 417	. . . . . . . . . . . 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 415	. . . . . . . . . 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 3090	. . . . . . . . . . . . 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 3459	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑣 ∈ V
46		eqeq1 2767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = 𝑣 → (𝑔 = {(𝑓‘𝑢), 𝑢} ↔ 𝑣 = {(𝑓‘𝑢), 𝑢}))
47	46	anbi2d 641	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = 𝑣 → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢})))
48	47	rexbidv 3189	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑣 → (∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) ↔ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢})))
49	45, 48	elab 3638	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ↔ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑣 = {(𝑓‘𝑢), 𝑢}))
50		neeq1 3020	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = 𝑢 → (𝑧 ≠ ∅ ↔ 𝑢 ≠ ∅))
51		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = 𝑢 → (𝑓‘𝑧) = (𝑓‘𝑢))
52	51	eleq1d 2848	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 = 𝑢 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑢) ∈ 𝑧))
53		eleq2 2852	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3578	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑢 ∈ 𝑥 → (𝑢 ≠ ∅ → (𝑓‘𝑢) ∈ 𝑢)))
57		elneq 9559	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 ∈ 𝑧 → 𝑤 ≠ 𝑧)
58	57	neneqd 2963	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 ∈ 𝑧 → ¬ 𝑤 = 𝑧)
59		vex 3459	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ 𝑤 ∈ V
60		neqne 2966	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑤 = 𝑧 → 𝑤 ≠ 𝑧)
61		prel12g 4829	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2852	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ {(𝑓‘𝑢), 𝑢}))
64		eleq2 2852	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → (𝑧 ∈ 𝑣 ↔ 𝑧 ∈ {(𝑓‘𝑢), 𝑢}))
65	63, 64	anbi12d 643	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑣 = {(𝑓‘𝑢), 𝑢} → ((𝑤 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣) ↔ (𝑤 ∈ {(𝑓‘𝑢), 𝑢} ∧ 𝑧 ∈ {(𝑓‘𝑢), 𝑢})))
66		ancom 465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 519	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑣 = {(𝑓‘𝑢), 𝑢} ∧ ¬ 𝑤 = 𝑧) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
69	58, 68	sylan2 604	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑣 = {(𝑓‘𝑢), 𝑢} ∧ 𝑤 ∈ 𝑧) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
70	69	adantrr 729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑣 = {(𝑓‘𝑢), 𝑢} ∧ (𝑤 ∈ 𝑧 ∧ (𝑓‘𝑢) ∈ 𝑢)) → ({𝑤, 𝑧} = {(𝑓‘𝑢), 𝑢} ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
71	70	pm5.32da 589	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9581	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = 𝑢 → (𝑤 = (𝑓‘𝑧) ↔ 𝑤 = (𝑓‘𝑢)))
75	74	biimparc 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 437	. . . . . . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . . 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 427	. . . . . . . . . . . . . . . . . . . 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 3159	. . . . . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . . . . 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 426	. . . . . . . . . . . . . 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 1963	. . . . . . . . . . . . 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 458	. . . . . . . . . . 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 1957	. . . . . . . . . 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 3677	. . . . . . . . . 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 535	. . . . . . . 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 3370	. . . . . . . . 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 2597	. . . . . . . . 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 420	. . . . . 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 3263	. . . . 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 3459	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
103	102	rnex 7903	. . . . . . . . . 10 ⊢ ran 𝑓 ∈ V
104		p0ex 5355	. . . . . . . . . 10 ⊢ {∅} ∈ V
105	103, 104	unex 7742	. . . . . . . . 9 ⊢ (ran 𝑓 ∪ {∅}) ∈ V
106		vex 3459	. . . . . . . . 9 ⊢ 𝑥 ∈ V
107	105, 106	unex 7742	. . . . . . . 8 ⊢ ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∈ V
108	107	pwex 5351	. . . . . . 7 ⊢ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∈ V
109		ssun1 4131	. . . . . . . . . . . . . 14 ⊢ (ran 𝑓 ∪ {∅}) ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
110		fvrn0 6909	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑢) ∈ (ran 𝑓 ∪ {∅})
111	109, 110	sselii 3934	. . . . . . . . . . . . 13 ⊢ (𝑓‘𝑢) ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
112		elun2 4136	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ 𝑥 → 𝑢 ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
113		prssi 4787	. . . . . . . . . . . . 13 ⊢ (((𝑓‘𝑢) ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥) ∧ 𝑢 ∈ ((ran 𝑓 ∪ {∅}) ∪ 𝑥)) → {(𝑓‘𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
114	111, 112, 113	sylancr 598	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝑥 → {(𝑓‘𝑢), 𝑢} ⊆ ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
115		prex 5409	. . . . . . . . . . . . 13 ⊢ {(𝑓‘𝑢), 𝑢} ∈ V
116	115	elpw 4566	. . . . . . . . . . . 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 2851	. . . . . . . . . . 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 495	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝑥 → ((𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)))
121	120	rexlimiv 3159	. . . . . . . 8 ⊢ (∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢}) → 𝑔 ∈ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥))
122	121	abssi 4022	. . . . . . 7 ⊢ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ⊆ 𝒫 ((ran 𝑓 ∪ {∅}) ∪ 𝑥)
123	108, 122	ssexi 5293	. . . . . 6 ⊢ {𝑔 ∣ ∃𝑢 ∈ 𝑥 (𝑢 ≠ ∅ ∧ 𝑔 = {(𝑓‘𝑢), 𝑢})} ∈ V
124		rexeq 3319	. . . . . . . . 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 3385	. . . . . . . 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 3188	. . . . . 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 3565	. . . . 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 1960	. . 3 ⊢ (∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
131	130	alimi 1841	. 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)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∃*wmo 2565  ∃!weu 2596  {cab 2741   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  ∃!wreu 3367  Vcvv 3455   ∪ cun 3903   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4562  {csn 4589  {cpr 4591  ran crn 5662  ‘cfv 6536  CHOICEwac 10095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-reg 9550

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-eprel 5561  df-fr 5614  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-ac 10096

This theorem is referenced by:  dfac2  10111