nnfoctbdjlem - Mathbox for Glauco Siliprandi

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Theorem nnfoctbdjlem 41578

Description: There exists a mapping from ℕ onto any (nonempty) countable set of disjoint sets, such that elements in the range of the map are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
nnfoctbdjlem.a	⊢ (𝜑 → 𝐴 ⊆ ℕ)
nnfoctbdjlem.g	⊢ (𝜑 → 𝐺:𝐴–1-1-onto→𝑋)
nnfoctbdjlem.dj	⊢ (𝜑 → Disj 𝑦 ∈ 𝑋 𝑦)
nnfoctbdjlem.f	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))))

**Assertion**
Ref	Expression
nnfoctbdjlem	⊢ (𝜑 → ∃𝑓(𝑓:ℕ–onto→(𝑋 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑓‘𝑛)))

Distinct variable groups: 𝐴,𝑛,𝑦 𝑓,𝐹,𝑛 𝑦,𝐹 𝑛,𝐺,𝑦 𝑓,𝑋,𝑛 𝑦,𝑋 𝜑,𝑛,𝑦 Allowed substitution hints: 𝜑(𝑓) 𝐴(𝑓) 𝐺(𝑓)

Proof of Theorem nnfoctbdjlem Dummy variables 𝑘 𝑥 𝑚 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iftrue 4312	. . . . . . . 8 ⊢ ((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = ∅)
2	1	adantl 475	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = ∅)
3		0ex 5026	. . . . . . . . 9 ⊢ ∅ ∈ V
4	3	snid 4429	. . . . . . . 8 ⊢ ∅ ∈ {∅}
5		elun2 4003	. . . . . . . 8 ⊢ (∅ ∈ {∅} → ∅ ∈ (𝑋 ∪ {∅}))
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ ∅ ∈ (𝑋 ∪ {∅})
7	2, 6	syl6eqel 2866	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) ∈ (𝑋 ∪ {∅}))
8	7	adantll 704	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) ∈ (𝑋 ∪ {∅}))
9		iffalse 4315	. . . . . . 7 ⊢ (¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = (𝐺‘(𝑛 − 1)))
10	9	adantl 475	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = (𝐺‘(𝑛 − 1)))
11		nnfoctbdjlem.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:𝐴–1-1-onto→𝑋)
12		f1of 6391	. . . . . . . . . . 11 ⊢ (𝐺:𝐴–1-1-onto→𝑋 → 𝐺:𝐴⟶𝑋)
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:𝐴⟶𝑋)
14	13	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → 𝐺:𝐴⟶𝑋)
15		pm2.46 869	. . . . . . . . . . 11 ⊢ (¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴) → ¬ ¬ (𝑛 − 1) ∈ 𝐴)
16	15	notnotrd 131	. . . . . . . . . 10 ⊢ (¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴) → (𝑛 − 1) ∈ 𝐴)
17	16	adantl 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → (𝑛 − 1) ∈ 𝐴)
18	14, 17	ffvelrnd 6624	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → (𝐺‘(𝑛 − 1)) ∈ 𝑋)
19	18	adantlr 705	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → (𝐺‘(𝑛 − 1)) ∈ 𝑋)
20		elun1 4002	. . . . . . 7 ⊢ ((𝐺‘(𝑛 − 1)) ∈ 𝑋 → (𝐺‘(𝑛 − 1)) ∈ (𝑋 ∪ {∅}))
21	19, 20	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → (𝐺‘(𝑛 − 1)) ∈ (𝑋 ∪ {∅}))
22	10, 21	eqeltrd 2858	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) ∈ (𝑋 ∪ {∅}))
23	8, 22	pm2.61dan 803	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) ∈ (𝑋 ∪ {∅}))
24		nnfoctbdjlem.f	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))))
25	23, 24	fmptd 6648	. . 3 ⊢ (𝜑 → 𝐹:ℕ⟶(𝑋 ∪ {∅}))
26		simpr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
27		f1ofo 6398	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐴–1-1-onto→𝑋 → 𝐺:𝐴–onto→𝑋)
28		forn 6369	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐴–onto→𝑋 → ran 𝐺 = 𝑋)
29	11, 27, 28	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐺 = 𝑋)
30	29	eqcomd 2783	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 = ran 𝐺)
31	30	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑋 = ran 𝐺)
32	26, 31	eleqtrd 2860	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ran 𝐺)
33	13	ffnd 6292	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 Fn 𝐴)
34		fvelrnb 6503	. . . . . . . . . . 11 ⊢ (𝐺 Fn 𝐴 → (𝑦 ∈ ran 𝐺 ↔ ∃𝑘 ∈ 𝐴 (𝐺‘𝑘) = 𝑦))
35	33, 34	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ ran 𝐺 ↔ ∃𝑘 ∈ 𝐴 (𝐺‘𝑘) = 𝑦))
36	35	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑦 ∈ ran 𝐺 ↔ ∃𝑘 ∈ 𝐴 (𝐺‘𝑘) = 𝑦))
37	32, 36	mpbid 224	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ∃𝑘 ∈ 𝐴 (𝐺‘𝑘) = 𝑦)
38		nnfoctbdjlem.a	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
39	38	sselda 3820	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℕ)
40	39	peano2nnd 11393	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 + 1) ∈ ℕ)
41	40	3adant3 1123	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) = 𝑦) → (𝑘 + 1) ∈ ℕ)
42	24	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐹 = (𝑛 ∈ ℕ ↦ if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1)))))
43		1red 10377	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → 1 ∈ ℝ)
44		1red 10377	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 1 ∈ ℝ)
45	39	nnrpd 12179	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℝ⁺)
46	44, 45	ltaddrp2d 12215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 1 < (𝑘 + 1))
47	46	adantr 474	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → 1 < (𝑘 + 1))
48		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = (𝑘 + 1) → 𝑛 = (𝑘 + 1))
49	48	eqcomd 2783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = (𝑘 + 1) → (𝑘 + 1) = 𝑛)
50	49	adantl 475	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → (𝑘 + 1) = 𝑛)
51	47, 50	breqtrd 4912	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → 1 < 𝑛)
52	43, 51	gtned 10511	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → 𝑛 ≠ 1)
53	52	neneqd 2973	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → ¬ 𝑛 = 1)
54		oveq1 6929	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 − 1) = ((𝑘 + 1) − 1))
55	39	nncnd 11392	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℂ)
56		1cnd 10371	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 1 ∈ ℂ)
57	55, 56	pncand 10735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 + 1) − 1) = 𝑘)
58	54, 57	sylan9eqr 2835	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → (𝑛 − 1) = 𝑘)
59		simplr 759	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → 𝑘 ∈ 𝐴)
60	58, 59	eqeltrd 2858	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → (𝑛 − 1) ∈ 𝐴)
61	60	notnotd 141	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → ¬ ¬ (𝑛 − 1) ∈ 𝐴)
62		ioran 969	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴) ↔ (¬ 𝑛 = 1 ∧ ¬ ¬ (𝑛 − 1) ∈ 𝐴))
63	53, 61, 62	sylanbrc 578	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴))
64	63	iffalsed 4317	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = (𝐺‘(𝑛 − 1)))
65	58	fveq2d 6450	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → (𝐺‘(𝑛 − 1)) = (𝐺‘𝑘))
66	64, 65	eqtrd 2813	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = (𝐺‘𝑘))
67	13	ffvelrnda 6623	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐺‘𝑘) ∈ 𝑋)
68	42, 66, 40, 67	fvmptd 6548	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘(𝑘 + 1)) = (𝐺‘𝑘))
69	68	3adant3 1123	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) = 𝑦) → (𝐹‘(𝑘 + 1)) = (𝐺‘𝑘))
70		simp3 1129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) = 𝑦) → (𝐺‘𝑘) = 𝑦)
71	69, 70	eqtrd 2813	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) = 𝑦) → (𝐹‘(𝑘 + 1)) = 𝑦)
72		fveq2 6446	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 + 1) → (𝐹‘𝑚) = (𝐹‘(𝑘 + 1)))
73	72	eqeq1d 2779	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → ((𝐹‘𝑚) = 𝑦 ↔ (𝐹‘(𝑘 + 1)) = 𝑦))
74	73	rspcev 3510	. . . . . . . . . . . 12 ⊢ (((𝑘 + 1) ∈ ℕ ∧ (𝐹‘(𝑘 + 1)) = 𝑦) → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = 𝑦)
75	41, 71, 74	syl2anc 579	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) = 𝑦) → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = 𝑦)
76	75	3exp 1109	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → ((𝐺‘𝑘) = 𝑦 → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = 𝑦)))
77	76	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑘 ∈ 𝐴 → ((𝐺‘𝑘) = 𝑦 → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = 𝑦)))
78	77	rexlimdv 3211	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (∃𝑘 ∈ 𝐴 (𝐺‘𝑘) = 𝑦 → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = 𝑦))
79	37, 78	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = 𝑦)
80		id 22	. . . . . . . . . 10 ⊢ ((𝐹‘𝑚) = 𝑦 → (𝐹‘𝑚) = 𝑦)
81	80	eqcomd 2783	. . . . . . . . 9 ⊢ ((𝐹‘𝑚) = 𝑦 → 𝑦 = (𝐹‘𝑚))
82	81	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑚 ∈ ℕ) → ((𝐹‘𝑚) = 𝑦 → 𝑦 = (𝐹‘𝑚)))
83	82	reximdva 3197	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (∃𝑚 ∈ ℕ (𝐹‘𝑚) = 𝑦 → ∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚)))
84	79, 83	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚))
85	84	adantlr 705	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋 ∪ {∅})) ∧ 𝑦 ∈ 𝑋) → ∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚))
86		simpll 757	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋 ∪ {∅})) ∧ ¬ 𝑦 ∈ 𝑋) → 𝜑)
87		elunnel1 3976	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝑋 ∪ {∅}) ∧ ¬ 𝑦 ∈ 𝑋) → 𝑦 ∈ {∅})
88		elsni 4414	. . . . . . . 8 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
89	87, 88	syl 17	. . . . . . 7 ⊢ ((𝑦 ∈ (𝑋 ∪ {∅}) ∧ ¬ 𝑦 ∈ 𝑋) → 𝑦 = ∅)
90	89	adantll 704	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋 ∪ {∅})) ∧ ¬ 𝑦 ∈ 𝑋) → 𝑦 = ∅)
91		1nn 11387	. . . . . . . 8 ⊢ 1 ∈ ℕ
92	91	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = ∅) → 1 ∈ ℕ)
93	1	orcs 864	. . . . . . . . . 10 ⊢ (𝑛 = 1 → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = ∅)
94	91	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℕ)
95	3	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ V)
96	24, 93, 94, 95	fvmptd3 6564	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1) = ∅)
97	96	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = ∅) → (𝐹‘1) = ∅)
98		id 22	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → 𝑦 = ∅)
99	98	eqcomd 2783	. . . . . . . . 9 ⊢ (𝑦 = ∅ → ∅ = 𝑦)
100	99	adantl 475	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = ∅) → ∅ = 𝑦)
101	97, 100	eqtr2d 2814	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = ∅) → 𝑦 = (𝐹‘1))
102		fveq2 6446	. . . . . . . 8 ⊢ (𝑚 = 1 → (𝐹‘𝑚) = (𝐹‘1))
103	102	rspceeqv 3528	. . . . . . 7 ⊢ ((1 ∈ ℕ ∧ 𝑦 = (𝐹‘1)) → ∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚))
104	92, 101, 103	syl2anc 579	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = ∅) → ∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚))
105	86, 90, 104	syl2anc 579	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋 ∪ {∅})) ∧ ¬ 𝑦 ∈ 𝑋) → ∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚))
106	85, 105	pm2.61dan 803	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∪ {∅})) → ∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚))
107	106	ralrimiva 3147	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (𝑋 ∪ {∅})∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚))
108		dffo3 6638	. . 3 ⊢ (𝐹:ℕ–onto→(𝑋 ∪ {∅}) ↔ (𝐹:ℕ⟶(𝑋 ∪ {∅}) ∧ ∀𝑦 ∈ (𝑋 ∪ {∅})∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚)))
109	25, 107, 108	sylanbrc 578	. 2 ⊢ (𝜑 → 𝐹:ℕ–onto→(𝑋 ∪ {∅}))
110		animorrl 966	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 = 𝑚) → (𝑛 = 𝑚 ∨ ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅))
111	2, 3	syl6eqel 2866	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) ∈ V)
112	24	fvmpt2 6552	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) ∈ V) → (𝐹‘𝑛) = if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))))
113	111, 112	syldan 585	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → (𝐹‘𝑛) = if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))))
114	113, 2	eqtrd 2813	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → (𝐹‘𝑛) = ∅)
115	114	ineq1d 4035	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = (∅ ∩ (𝐹‘𝑚)))
116		0in 4194	. . . . . . . . . 10 ⊢ (∅ ∩ (𝐹‘𝑚)) = ∅
117	115, 116	syl6eq 2829	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
118	117	adantlr 705	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
119	118	ad4ant24 744	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
120		eqeq1 2781	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (𝑛 = 1 ↔ 𝑚 = 1))
121		oveq1 6929	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (𝑛 − 1) = (𝑚 − 1))
122	121	eleq1d 2843	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → ((𝑛 − 1) ∈ 𝐴 ↔ (𝑚 − 1) ∈ 𝐴))
123	122	notbid 310	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (¬ (𝑛 − 1) ∈ 𝐴 ↔ ¬ (𝑚 − 1) ∈ 𝐴))
124	120, 123	orbi12d 905	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴) ↔ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)))
125	121	fveq2d 6450	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐺‘(𝑛 − 1)) = (𝐺‘(𝑚 − 1)))
126	124, 125	ifbieq2d 4331	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = if((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑚 − 1))))
127		simpl 476	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → 𝑚 ∈ ℕ)
128		iftrue 4312	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴) → if((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑚 − 1))) = ∅)
129	128, 3	syl6eqel 2866	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴) → if((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑚 − 1))) ∈ V)
130	129	adantl 475	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → if((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑚 − 1))) ∈ V)
131	24, 126, 127, 130	fvmptd3 6564	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐹‘𝑚) = if((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑚 − 1))))
132	128	adantl 475	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → if((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑚 − 1))) = ∅)
133	131, 132	eqtrd 2813	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐹‘𝑚) = ∅)
134	133	ineq2d 4036	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ((𝐹‘𝑛) ∩ ∅))
135		in0 4193	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑛) ∩ ∅) = ∅
136	134, 135	syl6eq 2829	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
137	136	adantll 704	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
138	137	ad5ant25 752	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
139		fvex 6459	. . . . . . . . . . . . . . . 16 ⊢ (𝐺‘(𝑛 − 1)) ∈ V
140	3, 139	ifex 4354	. . . . . . . . . . . . . . 15 ⊢ if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) ∈ V
141	140, 112	mpan2 681	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (𝐹‘𝑛) = if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))))
142	141, 9	sylan9eq 2833	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → (𝐹‘𝑛) = (𝐺‘(𝑛 − 1)))
143	142	adantlr 705	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → (𝐹‘𝑛) = (𝐺‘(𝑛 − 1)))
144	143	3adant3 1123	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐹‘𝑛) = (𝐺‘(𝑛 − 1)))
145	24	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → 𝐹 = (𝑛 ∈ ℕ ↦ if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1)))))
146	126	adantl 475	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) ∧ 𝑛 = 𝑚) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = if((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑚 − 1))))
147		iffalse 4315	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴) → if((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑚 − 1))) = (𝐺‘(𝑚 − 1)))
148	147	ad2antlr 717	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) ∧ 𝑛 = 𝑚) → if((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑚 − 1))) = (𝐺‘(𝑚 − 1)))
149	146, 148	eqtrd 2813	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) ∧ 𝑛 = 𝑚) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = (𝐺‘(𝑚 − 1)))
150		simpl 476	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → 𝑚 ∈ ℕ)
151		fvexd 6461	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐺‘(𝑚 − 1)) ∈ V)
152	145, 149, 150, 151	fvmptd 6548	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐹‘𝑚) = (𝐺‘(𝑚 − 1)))
153	152	adantll 704	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐹‘𝑚) = (𝐺‘(𝑚 − 1)))
154	153	3adant2 1122	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐹‘𝑚) = (𝐺‘(𝑚 − 1)))
155	144, 154	ineq12d 4037	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ((𝐺‘(𝑛 − 1)) ∩ (𝐺‘(𝑚 − 1))))
156	155	ad5ant245 1423	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ((𝐺‘(𝑛 − 1)) ∩ (𝐺‘(𝑚 − 1))))
157	16	ad2antlr 717	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝑛 − 1) ∈ 𝐴)
158		pm2.46 869	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴) → ¬ ¬ (𝑚 − 1) ∈ 𝐴)
159	158	notnotrd 131	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴) → (𝑚 − 1) ∈ 𝐴)
160	159	adantl 475	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝑚 − 1) ∈ 𝐴)
161		f1of1 6390	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺:𝐴–1-1-onto→𝑋 → 𝐺:𝐴–1-1→𝑋)
162	11, 161	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺:𝐴–1-1→𝑋)
163		dff14a 6799	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:𝐴–1-1→𝑋 ↔ (𝐺:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦))))
164	162, 163	sylib 210	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦))))
165	164	simprd 491	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦)))
166	165	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦)))
167	166	ad3antrrr 720	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦)))
168	157, 160, 167	jca31 510	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (((𝑛 − 1) ∈ 𝐴 ∧ (𝑚 − 1) ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦))))
169		nncn 11383	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
170	169	adantr 474	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → 𝑛 ∈ ℂ)
171	170	ad2antlr 717	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) → 𝑛 ∈ ℂ)
172		nncn 11383	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
173	172	adantl 475	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
174	173	ad2antlr 717	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) → 𝑚 ∈ ℂ)
175		1cnd 10371	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) → 1 ∈ ℂ)
176		simpr 479	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) → 𝑛 ≠ 𝑚)
177	171, 174, 175, 176	subneintr2d 10780	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) → (𝑛 − 1) ≠ (𝑚 − 1))
178	177	ad2antrr 716	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝑛 − 1) ≠ (𝑚 − 1))
179		neeq1 3030	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑛 − 1) → (𝑥 ≠ 𝑦 ↔ (𝑛 − 1) ≠ 𝑦))
180		fveq2 6446	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑛 − 1) → (𝐺‘𝑥) = (𝐺‘(𝑛 − 1)))
181	180	neeq1d 3027	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑛 − 1) → ((𝐺‘𝑥) ≠ (𝐺‘𝑦) ↔ (𝐺‘(𝑛 − 1)) ≠ (𝐺‘𝑦)))
182	179, 181	imbi12d 336	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑛 − 1) → ((𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦)) ↔ ((𝑛 − 1) ≠ 𝑦 → (𝐺‘(𝑛 − 1)) ≠ (𝐺‘𝑦))))
183		neeq2 3031	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑚 − 1) → ((𝑛 − 1) ≠ 𝑦 ↔ (𝑛 − 1) ≠ (𝑚 − 1)))
184		fveq2 6446	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑚 − 1) → (𝐺‘𝑦) = (𝐺‘(𝑚 − 1)))
185	184	neeq2d 3028	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑚 − 1) → ((𝐺‘(𝑛 − 1)) ≠ (𝐺‘𝑦) ↔ (𝐺‘(𝑛 − 1)) ≠ (𝐺‘(𝑚 − 1))))
186	183, 185	imbi12d 336	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑚 − 1) → (((𝑛 − 1) ≠ 𝑦 → (𝐺‘(𝑛 − 1)) ≠ (𝐺‘𝑦)) ↔ ((𝑛 − 1) ≠ (𝑚 − 1) → (𝐺‘(𝑛 − 1)) ≠ (𝐺‘(𝑚 − 1)))))
187	182, 186	rspc2va 3524	. . . . . . . . . . . 12 ⊢ ((((𝑛 − 1) ∈ 𝐴 ∧ (𝑚 − 1) ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦))) → ((𝑛 − 1) ≠ (𝑚 − 1) → (𝐺‘(𝑛 − 1)) ≠ (𝐺‘(𝑚 − 1))))
188	168, 178, 187	sylc 65	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐺‘(𝑛 − 1)) ≠ (𝐺‘(𝑚 − 1)))
189	188	neneqd 2973	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ¬ (𝐺‘(𝑛 − 1)) = (𝐺‘(𝑚 − 1)))
190	18	ad4ant13 741	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐺‘(𝑛 − 1)) ∈ 𝑋)
191	13	ffvelrnda 6623	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 − 1) ∈ 𝐴) → (𝐺‘(𝑚 − 1)) ∈ 𝑋)
192	159, 191	sylan2 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐺‘(𝑚 − 1)) ∈ 𝑋)
193	192	ad4ant14 742	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐺‘(𝑚 − 1)) ∈ 𝑋)
194		nnfoctbdjlem.dj	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Disj 𝑦 ∈ 𝑋 𝑦)
195		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
196	195	disjor 4868	. . . . . . . . . . . . . 14 ⊢ (Disj 𝑦 ∈ 𝑋 𝑦 ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦 = 𝑧 ∨ (𝑦 ∩ 𝑧) = ∅))
197	194, 196	sylib 210	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦 = 𝑧 ∨ (𝑦 ∩ 𝑧) = ∅))
198	197	ad3antrrr 720	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦 = 𝑧 ∨ (𝑦 ∩ 𝑧) = ∅))
199		eqeq1 2781	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐺‘(𝑛 − 1)) → (𝑦 = 𝑧 ↔ (𝐺‘(𝑛 − 1)) = 𝑧))
200		ineq1 4029	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝐺‘(𝑛 − 1)) → (𝑦 ∩ 𝑧) = ((𝐺‘(𝑛 − 1)) ∩ 𝑧))
201	200	eqeq1d 2779	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐺‘(𝑛 − 1)) → ((𝑦 ∩ 𝑧) = ∅ ↔ ((𝐺‘(𝑛 − 1)) ∩ 𝑧) = ∅))
202	199, 201	orbi12d 905	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐺‘(𝑛 − 1)) → ((𝑦 = 𝑧 ∨ (𝑦 ∩ 𝑧) = ∅) ↔ ((𝐺‘(𝑛 − 1)) = 𝑧 ∨ ((𝐺‘(𝑛 − 1)) ∩ 𝑧) = ∅)))
203		eqeq2 2788	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐺‘(𝑚 − 1)) → ((𝐺‘(𝑛 − 1)) = 𝑧 ↔ (𝐺‘(𝑛 − 1)) = (𝐺‘(𝑚 − 1))))
204		ineq2 4030	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐺‘(𝑚 − 1)) → ((𝐺‘(𝑛 − 1)) ∩ 𝑧) = ((𝐺‘(𝑛 − 1)) ∩ (𝐺‘(𝑚 − 1))))
205	204	eqeq1d 2779	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐺‘(𝑚 − 1)) → (((𝐺‘(𝑛 − 1)) ∩ 𝑧) = ∅ ↔ ((𝐺‘(𝑛 − 1)) ∩ (𝐺‘(𝑚 − 1))) = ∅))
206	203, 205	orbi12d 905	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐺‘(𝑚 − 1)) → (((𝐺‘(𝑛 − 1)) = 𝑧 ∨ ((𝐺‘(𝑛 − 1)) ∩ 𝑧) = ∅) ↔ ((𝐺‘(𝑛 − 1)) = (𝐺‘(𝑚 − 1)) ∨ ((𝐺‘(𝑛 − 1)) ∩ (𝐺‘(𝑚 − 1))) = ∅)))
207	202, 206	rspc2va 3524	. . . . . . . . . . . 12 ⊢ ((((𝐺‘(𝑛 − 1)) ∈ 𝑋 ∧ (𝐺‘(𝑚 − 1)) ∈ 𝑋) ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦 = 𝑧 ∨ (𝑦 ∩ 𝑧) = ∅)) → ((𝐺‘(𝑛 − 1)) = (𝐺‘(𝑚 − 1)) ∨ ((𝐺‘(𝑛 − 1)) ∩ (𝐺‘(𝑚 − 1))) = ∅))
208	190, 193, 198, 207	syl21anc 828	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ((𝐺‘(𝑛 − 1)) = (𝐺‘(𝑚 − 1)) ∨ ((𝐺‘(𝑛 − 1)) ∩ (𝐺‘(𝑚 − 1))) = ∅))
209	208	adantllr 709	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ((𝐺‘(𝑛 − 1)) = (𝐺‘(𝑚 − 1)) ∨ ((𝐺‘(𝑛 − 1)) ∩ (𝐺‘(𝑚 − 1))) = ∅))
210		orel1 875	. . . . . . . . . 10 ⊢ (¬ (𝐺‘(𝑛 − 1)) = (𝐺‘(𝑚 − 1)) → (((𝐺‘(𝑛 − 1)) = (𝐺‘(𝑚 − 1)) ∨ ((𝐺‘(𝑛 − 1)) ∩ (𝐺‘(𝑚 − 1))) = ∅) → ((𝐺‘(𝑛 − 1)) ∩ (𝐺‘(𝑚 − 1))) = ∅))
211	189, 209, 210	sylc 65	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ((𝐺‘(𝑛 − 1)) ∩ (𝐺‘(𝑚 − 1))) = ∅)
212	156, 211	eqtrd 2813	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
213	138, 212	pm2.61dan 803	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
214	119, 213	pm2.61dan 803	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
215	214	olcd 863	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) → (𝑛 = 𝑚 ∨ ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅))
216	110, 215	pm2.61dane 3056	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) → (𝑛 = 𝑚 ∨ ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅))
217	216	ralrimivva 3152	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑚 ∈ ℕ (𝑛 = 𝑚 ∨ ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅))
218		fveq2 6446	. . . 4 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
219	218	disjor 4868	. . 3 ⊢ (Disj 𝑛 ∈ ℕ (𝐹‘𝑛) ↔ ∀𝑛 ∈ ℕ ∀𝑚 ∈ ℕ (𝑛 = 𝑚 ∨ ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅))
220	217, 219	sylibr 226	. 2 ⊢ (𝜑 → Disj 𝑛 ∈ ℕ (𝐹‘𝑛))
221		nnex 11381	. . . . 5 ⊢ ℕ ∈ V
222	221	mptex 6758	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1)))) ∈ V
223	24, 222	eqeltri 2854	. . 3 ⊢ 𝐹 ∈ V
224		foeq1 6362	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:ℕ–onto→(𝑋 ∪ {∅}) ↔ 𝐹:ℕ–onto→(𝑋 ∪ {∅})))
225		simpl 476	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑛 ∈ ℕ) → 𝑓 = 𝐹)
226	225	fveq1d 6448	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) = (𝐹‘𝑛))
227	226	disjeq2dv 4859	. . . 4 ⊢ (𝑓 = 𝐹 → (Disj 𝑛 ∈ ℕ (𝑓‘𝑛) ↔ Disj 𝑛 ∈ ℕ (𝐹‘𝑛)))
228	224, 227	anbi12d 624	. . 3 ⊢ (𝑓 = 𝐹 → ((𝑓:ℕ–onto→(𝑋 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑓‘𝑛)) ↔ (𝐹:ℕ–onto→(𝑋 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝐹‘𝑛))))
229	223, 228	spcev 3501	. 2 ⊢ ((𝐹:ℕ–onto→(𝑋 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝐹‘𝑛)) → ∃𝑓(𝑓:ℕ–onto→(𝑋 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑓‘𝑛)))
230	109, 220, 229	syl2anc 579	1 ⊢ (𝜑 → ∃𝑓(𝑓:ℕ–onto→(𝑋 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑓‘𝑛)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 836 ∧ w3a 1071 = wceq 1601 ∃wex 1823 ∈ wcel 2106 ≠ wne 2968 ∀wral 3089 ∃wrex 3090 Vcvv 3397 ∪ cun 3789 ∩ cin 3790 ⊆ wss 3791 ∅c0 4140 ifcif 4306 {csn 4397 Disj wdisj 4854 class class class wbr 4886 ↦ cmpt 4965 ran crn 5356 Fn wfn 6130 ⟶wf 6131 –1-1→wf1 6132 –onto→wfo 6133 –1-1-onto→wf1o 6134 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 1c1 10273 + caddc 10275 < clt 10411 − cmin 10606 ℕcn 11374

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-disj 4855 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-rp 12138

This theorem is referenced by: nnfoctbdj 41579

W3C validator