nnfoctbdjlem - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > nnfoctbdjlem		Structured version Visualization version GIF version

Theorem nnfoctbdjlem 47227

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 4495	. . . . . . . 8 ⊢ ((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = ∅)
2	1	adantl 487	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = ∅)
3		0ex 5272	. . . . . . . . 9 ⊢ ∅ ∈ V
4	3	snid 4630	. . . . . . . 8 ⊢ ∅ ∈ {∅}
5		elun2 4136	. . . . . . . 8 ⊢ (∅ ∈ {∅} → ∅ ∈ (𝑋 ∪ {∅}))
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ ∅ ∈ (𝑋 ∪ {∅})
7	2, 6	eqeltrdi 2873	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) ∈ (𝑋 ∪ {∅}))
8	7	adantll 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) ∈ (𝑋 ∪ {∅}))
9		iffalse 4498	. . . . . . 7 ⊢ (¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = (𝐺‘(𝑛 − 1)))
10	9	adantl 487	. . . . . 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 6824	. . . . . . . . . . 11 ⊢ (𝐺:𝐴–1-1-onto→𝑋 → 𝐺:𝐴⟶𝑋)
13	11, 12	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:𝐴⟶𝑋)
14	13	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → 𝐺:𝐴⟶𝑋)
15		pm2.46 896	. . . . . . . . . . 11 ⊢ (¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴) → ¬ ¬ (𝑛 − 1) ∈ 𝐴)
16	15	notnotrd 134	. . . . . . . . . 10 ⊢ (¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴) → (𝑛 − 1) ∈ 𝐴)
17	16	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → (𝑛 − 1) ∈ 𝐴)
18	14, 17	ffvelcdmd 7084	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → (𝐺‘(𝑛 − 1)) ∈ 𝑋)
19	18	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → (𝐺‘(𝑛 − 1)) ∈ 𝑋)
20		elun1 4135	. . . . . . 7 ⊢ ((𝐺‘(𝑛 − 1)) ∈ 𝑋 → (𝐺‘(𝑛 − 1)) ∈ (𝑋 ∪ {∅}))
21	19, 20	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → (𝐺‘(𝑛 − 1)) ∈ (𝑋 ∪ {∅}))
22	10, 21	eqeltrd 2865	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) ∈ (𝑋 ∪ {∅}))
23	8, 22	pm2.61dan 825	. . . 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 7113	. . 3 ⊢ (𝜑 → 𝐹:ℕ⟶(𝑋 ∪ {∅}))
26		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
27		f1ofo 6832	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐴–1-1-onto→𝑋 → 𝐺:𝐴–onto→𝑋)
28		forn 6799	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐴–onto→𝑋 → ran 𝐺 = 𝑋)
29	11, 27, 28	3syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐺 = 𝑋)
30	29	eqcomd 2771	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 = ran 𝐺)
31	30	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑋 = ran 𝐺)
32	26, 31	eleqtrd 2867	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ran 𝐺)
33	13	ffnd 6710	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 Fn 𝐴)
34		fvelrnb 6945	. . . . . . . . . . 11 ⊢ (𝐺 Fn 𝐴 → (𝑦 ∈ ran 𝐺 ↔ ∃𝑘 ∈ 𝐴 (𝐺‘𝑘) = 𝑦))
35	33, 34	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ ran 𝐺 ↔ ∃𝑘 ∈ 𝐴 (𝐺‘𝑘) = 𝑦))
36	35	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑦 ∈ ran 𝐺 ↔ ∃𝑘 ∈ 𝐴 (𝐺‘𝑘) = 𝑦))
37	32, 36	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ∃𝑘 ∈ 𝐴 (𝐺‘𝑘) = 𝑦)
38		nnfoctbdjlem.a	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
39	38	sselda 3938	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℕ)
40	39	peano2nnd 12265	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 + 1) ∈ ℕ)
41	40	3adant3 1150	. . . . . . . . . . . 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 11224	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → 1 ∈ ℝ)
44		1red 11224	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 1 ∈ ℝ)
45	39	nnrpd 13074	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℝ⁺)
46	44, 45	ltaddrp2d 13110	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 1 < (𝑘 + 1))
47	46	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → 1 < (𝑘 + 1))
48		id 23	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = (𝑘 + 1) → 𝑛 = (𝑘 + 1))
49	48	eqcomd 2771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = (𝑘 + 1) → (𝑘 + 1) = 𝑛)
50	49	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → (𝑘 + 1) = 𝑛)
51	47, 50	breqtrd 5139	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → 1 < 𝑛)
52	43, 51	gtned 11360	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → 𝑛 ≠ 1)
53	52	neneqd 2965	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → ¬ 𝑛 = 1)
54		oveq1 7426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 − 1) = ((𝑘 + 1) − 1))
55	39	nncnd 12264	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℂ)
56		1cnd 11217	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 1 ∈ ℂ)
57	55, 56	pncand 11585	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 + 1) − 1) = 𝑘)
58	54, 57	sylan9eqr 2822	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → (𝑛 − 1) = 𝑘)
59		simplr 781	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → 𝑘 ∈ 𝐴)
60	58, 59	eqeltrd 2865	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → (𝑛 − 1) ∈ 𝐴)
61	60	notnotd 145	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → ¬ ¬ (𝑛 − 1) ∈ 𝐴)
62		ioran 999	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴) ↔ (¬ 𝑛 = 1 ∧ ¬ ¬ (𝑛 − 1) ∈ 𝐴))
63	53, 61, 62	sylanbrc 595	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴))
64	63	iffalsed 4500	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = (𝐺‘(𝑛 − 1)))
65	58	fveq2d 6889	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → (𝐺‘(𝑛 − 1)) = (𝐺‘𝑘))
66	64, 65	eqtrd 2800	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 = (𝑘 + 1)) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = (𝐺‘𝑘))
67	13	ffvelcdmda 7083	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐺‘𝑘) ∈ 𝑋)
68	42, 66, 40, 67	fvmptd 7001	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘(𝑘 + 1)) = (𝐺‘𝑘))
69	68	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) = 𝑦) → (𝐹‘(𝑘 + 1)) = (𝐺‘𝑘))
70		simp3 1156	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) = 𝑦) → (𝐺‘𝑘) = 𝑦)
71	69, 70	eqtrd 2800	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) = 𝑦) → (𝐹‘(𝑘 + 1)) = 𝑦)
72		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 + 1) → (𝐹‘𝑚) = (𝐹‘(𝑘 + 1)))
73	72	eqeq1d 2767	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → ((𝐹‘𝑚) = 𝑦 ↔ (𝐹‘(𝑘 + 1)) = 𝑦))
74	73	rspcev 3583	. . . . . . . . . . . 12 ⊢ (((𝑘 + 1) ∈ ℕ ∧ (𝐹‘(𝑘 + 1)) = 𝑦) → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = 𝑦)
75	41, 71, 74	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) = 𝑦) → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = 𝑦)
76	75	3exp 1137	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → ((𝐺‘𝑘) = 𝑦 → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = 𝑦)))
77	76	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑘 ∈ 𝐴 → ((𝐺‘𝑘) = 𝑦 → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = 𝑦)))
78	77	rexlimdv 3166	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (∃𝑘 ∈ 𝐴 (𝐺‘𝑘) = 𝑦 → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = 𝑦))
79	37, 78	mpd 16	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = 𝑦)
80		id 23	. . . . . . . . . 10 ⊢ ((𝐹‘𝑚) = 𝑦 → (𝐹‘𝑚) = 𝑦)
81	80	eqcomd 2771	. . . . . . . . 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 3180	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (∃𝑚 ∈ ℕ (𝐹‘𝑚) = 𝑦 → ∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚)))
84	79, 83	mpd 16	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚))
85	84	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋 ∪ {∅})) ∧ 𝑦 ∈ 𝑋) → ∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚))
86		simpll 779	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋 ∪ {∅})) ∧ ¬ 𝑦 ∈ 𝑋) → 𝜑)
87		elunnel1 4108	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝑋 ∪ {∅}) ∧ ¬ 𝑦 ∈ 𝑋) → 𝑦 ∈ {∅})
88		elsni 4608	. . . . . . . 8 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
89	87, 88	syl 18	. . . . . . 7 ⊢ ((𝑦 ∈ (𝑋 ∪ {∅}) ∧ ¬ 𝑦 ∈ 𝑋) → 𝑦 = ∅)
90	89	adantll 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋 ∪ {∅})) ∧ ¬ 𝑦 ∈ 𝑋) → 𝑦 = ∅)
91		1nn 12259	. . . . . . . 8 ⊢ 1 ∈ ℕ
92	91	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = ∅) → 1 ∈ ℕ)
93	1	orcs 889	. . . . . . . . . 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 7017	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1) = ∅)
97	96	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = ∅) → (𝐹‘1) = ∅)
98		id 23	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → 𝑦 = ∅)
99	98	eqcomd 2771	. . . . . . . . 9 ⊢ (𝑦 = ∅ → ∅ = 𝑦)
100	99	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = ∅) → ∅ = 𝑦)
101	97, 100	eqtr2d 2801	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = ∅) → 𝑦 = (𝐹‘1))
102		fveq2 6885	. . . . . . . 8 ⊢ (𝑚 = 1 → (𝐹‘𝑚) = (𝐹‘1))
103	102	rspceeqv 3606	. . . . . . 7 ⊢ ((1 ∈ ℕ ∧ 𝑦 = (𝐹‘1)) → ∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚))
104	92, 101, 103	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = ∅) → ∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚))
105	86, 90, 104	syl2anc 596	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋 ∪ {∅})) ∧ ¬ 𝑦 ∈ 𝑋) → ∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚))
106	85, 105	pm2.61dan 825	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∪ {∅})) → ∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚))
107	106	ralrimiva 3159	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (𝑋 ∪ {∅})∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚))
108		dffo3 7101	. . 3 ⊢ (𝐹:ℕ–onto→(𝑋 ∪ {∅}) ↔ (𝐹:ℕ⟶(𝑋 ∪ {∅}) ∧ ∀𝑦 ∈ (𝑋 ∪ {∅})∃𝑚 ∈ ℕ 𝑦 = (𝐹‘𝑚)))
109	25, 107, 108	sylanbrc 595	. 2 ⊢ (𝜑 → 𝐹:ℕ–onto→(𝑋 ∪ {∅}))
110		animorrl 996	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 = 𝑚) → (𝑛 = 𝑚 ∨ ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅))
111	2, 3	eqeltrdi 2873	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) ∈ V)
112	24	fvmpt2 7005	. . . . . . . . . . . . 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 603	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → (𝐹‘𝑛) = if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))))
114	113, 2	eqtrd 2800	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → (𝐹‘𝑛) = ∅)
115	114	ineq1d 4172	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = (∅ ∩ (𝐹‘𝑚)))
116		0in 4354	. . . . . . . . . 10 ⊢ (∅ ∩ (𝐹‘𝑚)) = ∅
117	115, 116	eqtrdi 2816	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
118	117	adantlr 728	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
119	118	ad4ant24 767	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
120		eqeq1 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (𝑛 = 1 ↔ 𝑚 = 1))
121		oveq1 7426	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (𝑛 − 1) = (𝑚 − 1))
122	121	eleq1d 2850	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → ((𝑛 − 1) ∈ 𝐴 ↔ (𝑚 − 1) ∈ 𝐴))
123	122	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (¬ (𝑛 − 1) ∈ 𝐴 ↔ ¬ (𝑚 − 1) ∈ 𝐴))
124	120, 123	orbi12d 932	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴) ↔ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)))
125	121	fveq2d 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐺‘(𝑛 − 1)) = (𝐺‘(𝑚 − 1)))
126	124, 125	ifbieq2d 4516	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = if((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑚 − 1))))
127		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → 𝑚 ∈ ℕ)
128		iftrue 4495	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴) → if((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑚 − 1))) = ∅)
129	128, 3	eqeltrdi 2873	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴) → if((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑚 − 1))) ∈ V)
130	129	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → if((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑚 − 1))) ∈ V)
131	24, 126, 127, 130	fvmptd3 7017	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐹‘𝑚) = if((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑚 − 1))))
132	128	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → if((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑚 − 1))) = ∅)
133	131, 132	eqtrd 2800	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐹‘𝑚) = ∅)
134	133	ineq2d 4173	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ((𝐹‘𝑛) ∩ ∅))
135		in0 4352	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑛) ∩ ∅) = ∅
136	134, 135	eqtrdi 2816	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
137	136	adantll 727	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
138	137	ad5ant25 774	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
139		fvex 6898	. . . . . . . . . . . . . . . 16 ⊢ (𝐺‘(𝑛 − 1)) ∈ V
140	3, 139	ifex 4540	. . . . . . . . . . . . . . 15 ⊢ if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) ∈ V
141	140, 112	mpan2 704	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (𝐹‘𝑛) = if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))))
142	141, 9	sylan9eq 2820	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → (𝐹‘𝑛) = (𝐺‘(𝑛 − 1)))
143	142	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → (𝐹‘𝑛) = (𝐺‘(𝑛 − 1)))
144	143	3adant3 1150	. . . . . . . . . . 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 487	. . . . . . . . . . . . . . 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 4498	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴) → if((𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑚 − 1))) = (𝐺‘(𝑚 − 1)))
148	147	ad2antlr 740	. . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) ∧ 𝑛 = 𝑚) → if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))) = (𝐺‘(𝑚 − 1)))
150		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → 𝑚 ∈ ℕ)
151		fvexd 6900	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐺‘(𝑚 − 1)) ∈ V)
152	145, 149, 150, 151	fvmptd 7001	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐹‘𝑚) = (𝐺‘(𝑚 − 1)))
153	152	adantll 727	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐹‘𝑚) = (𝐺‘(𝑚 − 1)))
154	153	3adant2 1149	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐹‘𝑚) = (𝐺‘(𝑚 − 1)))
155	144, 154	ineq12d 4174	. . . . . . . . . 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 1384	. . . . . . . . 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 740	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝑛 − 1) ∈ 𝐴)
158		pm2.46 896	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴) → ¬ ¬ (𝑚 − 1) ∈ 𝐴)
159	158	notnotrd 134	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴) → (𝑚 − 1) ∈ 𝐴)
160	159	adantl 487	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝑚 − 1) ∈ 𝐴)
161		f1of1 6823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺:𝐴–1-1-onto→𝑋 → 𝐺:𝐴–1-1→𝑋)
162	11, 161	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺:𝐴–1-1→𝑋)
163		dff14a 7273	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:𝐴–1-1→𝑋 ↔ (𝐺:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦))))
164	162, 163	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦))))
165	164	simprd 501	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦)))
166	165	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦)))
167	166	ad3antrrr 743	. . . . . . . . . . . . 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 524	. . . . . . . . . . . 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 12256	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
170	169	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → 𝑛 ∈ ℂ)
171	170	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) → 𝑛 ∈ ℂ)
172		nncn 12256	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
173	172	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
174	173	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) → 𝑚 ∈ ℂ)
175		1cnd 11217	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) → 1 ∈ ℂ)
176		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) → 𝑛 ≠ 𝑚)
177	171, 174, 175, 176	subneintr2d 11630	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) → (𝑛 − 1) ≠ (𝑚 − 1))
178	177	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝑛 − 1) ≠ (𝑚 − 1))
179		neeq1 3022	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑛 − 1) → (𝑥 ≠ 𝑦 ↔ (𝑛 − 1) ≠ 𝑦))
180		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑛 − 1) → (𝐺‘𝑥) = (𝐺‘(𝑛 − 1)))
181	180	neeq1d 3019	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑛 − 1) → ((𝐺‘𝑥) ≠ (𝐺‘𝑦) ↔ (𝐺‘(𝑛 − 1)) ≠ (𝐺‘𝑦)))
182	179, 181	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑛 − 1) → ((𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦)) ↔ ((𝑛 − 1) ≠ 𝑦 → (𝐺‘(𝑛 − 1)) ≠ (𝐺‘𝑦))))
183		neeq2 3023	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑚 − 1) → ((𝑛 − 1) ≠ 𝑦 ↔ (𝑛 − 1) ≠ (𝑚 − 1)))
184		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑚 − 1) → (𝐺‘𝑦) = (𝐺‘(𝑚 − 1)))
185	184	neeq2d 3020	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑚 − 1) → ((𝐺‘(𝑛 − 1)) ≠ (𝐺‘𝑦) ↔ (𝐺‘(𝑛 − 1)) ≠ (𝐺‘(𝑚 − 1))))
186	183, 185	imbi12d 347	. . . . . . . . . . . . 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 3595	. . . . . . . . . . . 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 66	. . . . . . . . . . 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 2965	. . . . . . . . . 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 764	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐺‘(𝑛 − 1)) ∈ 𝑋)
191	13	ffvelcdmda 7083	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 − 1) ∈ 𝐴) → (𝐺‘(𝑚 − 1)) ∈ 𝑋)
192	159, 191	sylan2 605	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑚 = 1 ∨ ¬ (𝑚 − 1) ∈ 𝐴)) → (𝐺‘(𝑚 − 1)) ∈ 𝑋)
193	192	ad4ant14 765	. . . . . . . . . . . 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 23	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
196	195	disjor 5093	. . . . . . . . . . . . . 14 ⊢ (Disj 𝑦 ∈ 𝑋 𝑦 ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦 = 𝑧 ∨ (𝑦 ∩ 𝑧) = ∅))
197	194, 196	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦 = 𝑧 ∨ (𝑦 ∩ 𝑧) = ∅))
198	197	ad3antrrr 743	. . . . . . . . . . . 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 2769	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐺‘(𝑛 − 1)) → (𝑦 = 𝑧 ↔ (𝐺‘(𝑛 − 1)) = 𝑧))
200		ineq1 4166	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝐺‘(𝑛 − 1)) → (𝑦 ∩ 𝑧) = ((𝐺‘(𝑛 − 1)) ∩ 𝑧))
201	200	eqeq1d 2767	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐺‘(𝑛 − 1)) → ((𝑦 ∩ 𝑧) = ∅ ↔ ((𝐺‘(𝑛 − 1)) ∩ 𝑧) = ∅))
202	199, 201	orbi12d 932	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐺‘(𝑛 − 1)) → ((𝑦 = 𝑧 ∨ (𝑦 ∩ 𝑧) = ∅) ↔ ((𝐺‘(𝑛 − 1)) = 𝑧 ∨ ((𝐺‘(𝑛 − 1)) ∩ 𝑧) = ∅)))
203		eqeq2 2777	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐺‘(𝑚 − 1)) → ((𝐺‘(𝑛 − 1)) = 𝑧 ↔ (𝐺‘(𝑛 − 1)) = (𝐺‘(𝑚 − 1))))
204		ineq2 4167	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐺‘(𝑚 − 1)) → ((𝐺‘(𝑛 − 1)) ∩ 𝑧) = ((𝐺‘(𝑛 − 1)) ∩ (𝐺‘(𝑚 − 1))))
205	204	eqeq1d 2767	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐺‘(𝑚 − 1)) → (((𝐺‘(𝑛 − 1)) ∩ 𝑧) = ∅ ↔ ((𝐺‘(𝑛 − 1)) ∩ (𝐺‘(𝑚 − 1))) = ∅))
206	203, 205	orbi12d 932	. . . . . . . . . . . . 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 3595	. . . . . . . . . . . 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 851	. . . . . . . . . . 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 732	. . . . . . . . . 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 902	. . . . . . . . . 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 66	. . . . . . . . 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 2800	. . . . . . . 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 825	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) ∧ ¬ (𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴)) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
214	119, 213	pm2.61dan 825	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) → ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅)
215	214	olcd 888	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑛 ≠ 𝑚) → (𝑛 = 𝑚 ∨ ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅))
216	110, 215	pm2.61dane 3047	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) → (𝑛 = 𝑚 ∨ ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅))
217	216	ralrimivva 3210	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑚 ∈ ℕ (𝑛 = 𝑚 ∨ ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅))
218		fveq2 6885	. . . 4 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
219	218	disjor 5093	. . 3 ⊢ (Disj 𝑛 ∈ ℕ (𝐹‘𝑛) ↔ ∀𝑛 ∈ ℕ ∀𝑚 ∈ ℕ (𝑛 = 𝑚 ∨ ((𝐹‘𝑛) ∩ (𝐹‘𝑚)) = ∅))
220	217, 219	sylibr 237	. 2 ⊢ (𝜑 → Disj 𝑛 ∈ ℕ (𝐹‘𝑛))
221		nnex 12254	. . . . 5 ⊢ ℕ ∈ V
222	221	mptex 7228	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1)))) ∈ V
223	24, 222	eqeltri 2861	. . 3 ⊢ 𝐹 ∈ V
224		foeq1 6792	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:ℕ–onto→(𝑋 ∪ {∅}) ↔ 𝐹:ℕ–onto→(𝑋 ∪ {∅})))
225		simpl 488	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑛 ∈ ℕ) → 𝑓 = 𝐹)
226	225	fveq1d 6887	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) = (𝐹‘𝑛))
227	226	disjeq2dv 5083	. . . 4 ⊢ (𝑓 = 𝐹 → (Disj 𝑛 ∈ ℕ (𝑓‘𝑛) ↔ Disj 𝑛 ∈ ℕ (𝐹‘𝑛)))
228	224, 227	anbi12d 644	. . 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 3567	. 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 596	1 ⊢ (𝜑 → ∃𝑓(𝑓:ℕ–onto→(𝑋 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑓‘𝑛)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∃wex 1812 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 ∃wrex 3091 Vcvv 3457 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∅c0 4286 ifcif 4489 {csn 4591 Disj wdisj 5078 class class class wbr 5111 ↦ cmpt 5194 ran crn 5664 Fn wfn 6535 ⟶wf 6536 –1-1→wf1 6537 –onto→wfo 6538 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7419 ℂcc 11113 1c1 11116 + caddc 11118 < clt 11258 − cmin 11456 ℕcn 12248

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-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 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-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-disj 5079 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 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-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-nn 12249 df-rp 13033

This theorem is used by: nnfoctbdj 47228

W3C validator