Theorem ctinfomlemom 12382

Description: Lemma for ctinfom 12383. Converting between ω and ℕ₀. (Contributed by Jim Kingdon, 10-Aug-2023.)

Hypotheses

Ref	Expression
ctinfom.n	⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
ctinfom.g	⊢ 𝐺 = (𝐹 ∘ ◡𝑁)
ctinfom.f	⊢ (𝜑 → 𝐹:ω–onto→𝐴)
ctinfom.inf	⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑛))

Assertion

Ref	Expression
ctinfomlemom	⊢ (𝜑 → (𝐺:ℕ0–onto→𝐴 ∧ ∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝐺‘𝑗) ≠ (𝐺‘𝑖)))

Distinct variable groups:   𝑖,𝐹,𝑥   𝑛,𝐹   𝑗,𝐺,𝑘   𝑖,𝑁,𝑗,𝑘   𝑛,𝑁,𝑘   𝑥,𝑁,𝑘   𝑖,𝑚,𝑗,𝑘   𝜑,𝑖,𝑘,𝑚,𝑥   𝑚,𝑛

Allowed substitution hints:   𝜑(𝑗,𝑛)   𝐴(𝑥,𝑖,𝑗,𝑘,𝑚,𝑛)   𝐹(𝑗,𝑘,𝑚)   𝐺(𝑥,𝑖,𝑚,𝑛)   𝑁(𝑚)

Proof of Theorem ctinfomlemom

Step	Hyp	Ref	Expression
1		ctinfom.f	. . . 4 ⊢ (𝜑 → 𝐹:ω–onto→𝐴)
2		ctinfom.n	. . . . . . 7 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
3	2	frechashgf1o 10384	. . . . . 6 ⊢ 𝑁:ω–1-1-onto→ℕ0
4		f1ocnv 5455	. . . . . 6 ⊢ (𝑁:ω–1-1-onto→ℕ0 → ◡𝑁:ℕ0–1-1-onto→ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ ◡𝑁:ℕ0–1-1-onto→ω
6		f1ofo 5449	. . . . 5 ⊢ (◡𝑁:ℕ0–1-1-onto→ω → ◡𝑁:ℕ0–onto→ω)
7	5, 6	ax-mp 5	. . . 4 ⊢ ◡𝑁:ℕ0–onto→ω
8		foco 5430	. . . 4 ⊢ ((𝐹:ω–onto→𝐴 ∧ ◡𝑁:ℕ0–onto→ω) → (𝐹 ∘ ◡𝑁):ℕ0–onto→𝐴)
9	1, 7, 8	sylancl 411	. . 3 ⊢ (𝜑 → (𝐹 ∘ ◡𝑁):ℕ0–onto→𝐴)
10		ctinfom.g	. . . 4 ⊢ 𝐺 = (𝐹 ∘ ◡𝑁)
11		foeq1 5416	. . . 4 ⊢ (𝐺 = (𝐹 ∘ ◡𝑁) → (𝐺:ℕ0–onto→𝐴 ↔ (𝐹 ∘ ◡𝑁):ℕ0–onto→𝐴))
12	10, 11	ax-mp 5	. . 3 ⊢ (𝐺:ℕ0–onto→𝐴 ↔ (𝐹 ∘ ◡𝑁):ℕ0–onto→𝐴)
13	9, 12	sylibr 133	. 2 ⊢ (𝜑 → 𝐺:ℕ0–onto→𝐴)
14		imaeq2 4949	. . . . . . . 8 ⊢ (𝑛 = suc (◡𝑁‘𝑚) → (𝐹 “ 𝑛) = (𝐹 “ suc (◡𝑁‘𝑚)))
15	14	eleq2d 2240	. . . . . . 7 ⊢ (𝑛 = suc (◡𝑁‘𝑚) → ((𝐹‘𝑘) ∈ (𝐹 “ 𝑛) ↔ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚))))
16	15	notbid 662	. . . . . 6 ⊢ (𝑛 = suc (◡𝑁‘𝑚) → (¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑛) ↔ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚))))
17	16	rexbidv 2471	. . . . 5 ⊢ (𝑛 = suc (◡𝑁‘𝑚) → (∃𝑘 ∈ ω ¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑛) ↔ ∃𝑘 ∈ ω ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚))))
18		ctinfom.inf	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑛))
19	18	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑛))
20		f1of 5442	. . . . . . . . 9 ⊢ (◡𝑁:ℕ0–1-1-onto→ω → ◡𝑁:ℕ0⟶ω)
21	5, 20	ax-mp 5	. . . . . . . 8 ⊢ ◡𝑁:ℕ0⟶ω
22	21	a1i 9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ◡𝑁:ℕ0⟶ω)
23		simpr 109	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
24	22, 23	ffvelrnd 5632	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (◡𝑁‘𝑚) ∈ ω)
25		peano2 4579	. . . . . 6 ⊢ ((◡𝑁‘𝑚) ∈ ω → suc (◡𝑁‘𝑚) ∈ ω)
26	24, 25	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → suc (◡𝑁‘𝑚) ∈ ω)
27	17, 19, 26	rspcdva 2839	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ∃𝑘 ∈ ω ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))
28		f1of 5442	. . . . . . . 8 ⊢ (𝑁:ω–1-1-onto→ℕ0 → 𝑁:ω⟶ℕ0)
29	3, 28	ax-mp 5	. . . . . . 7 ⊢ 𝑁:ω⟶ℕ0
30	29	a1i 9	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) → 𝑁:ω⟶ℕ0)
31		simprl 526	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) → 𝑘 ∈ ω)
32	30, 31	ffvelrnd 5632	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) → (𝑁‘𝑘) ∈ ℕ0)
33	10	fveq1i 5497	. . . . . . . . . . 11 ⊢ (𝐺‘(𝑁‘𝑘)) = ((𝐹 ∘ ◡𝑁)‘(𝑁‘𝑘))
34	32	adantr 274	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁‘𝑘) ∈ ℕ0)
35		fvco3 5567	. . . . . . . . . . . 12 ⊢ ((◡𝑁:ℕ0⟶ω ∧ (𝑁‘𝑘) ∈ ℕ0) → ((𝐹 ∘ ◡𝑁)‘(𝑁‘𝑘)) = (𝐹‘(◡𝑁‘(𝑁‘𝑘))))
36	21, 34, 35	sylancr 412	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((𝐹 ∘ ◡𝑁)‘(𝑁‘𝑘)) = (𝐹‘(◡𝑁‘(𝑁‘𝑘))))
37	33, 36	eqtrid 2215	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘(𝑁‘𝑘)) = (𝐹‘(◡𝑁‘(𝑁‘𝑘))))
38	31	adantr 274	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑘 ∈ ω)
39		f1ocnvfv1 5756	. . . . . . . . . . . . 13 ⊢ ((𝑁:ω–1-1-onto→ℕ0 ∧ 𝑘 ∈ ω) → (◡𝑁‘(𝑁‘𝑘)) = 𝑘)
40	3, 39	mpan 422	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ω → (◡𝑁‘(𝑁‘𝑘)) = 𝑘)
41	40	fveq2d 5500	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ω → (𝐹‘(◡𝑁‘(𝑁‘𝑘))) = (𝐹‘𝑘))
42	38, 41	syl 14	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐹‘(◡𝑁‘(𝑁‘𝑘))) = (𝐹‘𝑘))
43	37, 42	eqtrd 2203	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘(𝑁‘𝑘)) = (𝐹‘𝑘))
44		simplrr 531	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))
45	43, 44	eqneltrd 2266	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ¬ (𝐺‘(𝑁‘𝑘)) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))
46		simpr 109	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) ∧ (𝐺‘(𝑁‘𝑘)) = (𝐺‘𝑖)) → (𝐺‘(𝑁‘𝑘)) = (𝐺‘𝑖))
47	10	fveq1i 5497	. . . . . . . . . . . 12 ⊢ (𝐺‘𝑖) = ((𝐹 ∘ ◡𝑁)‘𝑖)
48		elfznn0 10070	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0...𝑚) → 𝑖 ∈ ℕ0)
49	48	adantl 275	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑖 ∈ ℕ0)
50		fvco3 5567	. . . . . . . . . . . . 13 ⊢ ((◡𝑁:ℕ0⟶ω ∧ 𝑖 ∈ ℕ0) → ((𝐹 ∘ ◡𝑁)‘𝑖) = (𝐹‘(◡𝑁‘𝑖)))
51	21, 49, 50	sylancr 412	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((𝐹 ∘ ◡𝑁)‘𝑖) = (𝐹‘(◡𝑁‘𝑖)))
52	47, 51	eqtrid 2215	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘𝑖) = (𝐹‘(◡𝑁‘𝑖)))
53		elfzle2 9984	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0...𝑚) → 𝑖 ≤ 𝑚)
54	53	adantl 275	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑖 ≤ 𝑚)
55		0zd 9224	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 0 ∈ ℤ)
56	21	a1i 9	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ◡𝑁:ℕ0⟶ω)
57	56, 49	ffvelrnd 5632	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (◡𝑁‘𝑖) ∈ ω)
58	24	ad2antrr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (◡𝑁‘𝑚) ∈ ω)
59	55, 2, 57, 58	frec2uzled 10385	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((◡𝑁‘𝑖) ⊆ (◡𝑁‘𝑚) ↔ (𝑁‘(◡𝑁‘𝑖)) ≤ (𝑁‘(◡𝑁‘𝑚))))
60		f1ocnvfv2 5757	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁:ω–1-1-onto→ℕ0 ∧ 𝑖 ∈ ℕ0) → (𝑁‘(◡𝑁‘𝑖)) = 𝑖)
61	3, 49, 60	sylancr 412	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁‘(◡𝑁‘𝑖)) = 𝑖)
62	23	ad2antrr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑚 ∈ ℕ0)
63		f1ocnvfv2 5757	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁:ω–1-1-onto→ℕ0 ∧ 𝑚 ∈ ℕ0) → (𝑁‘(◡𝑁‘𝑚)) = 𝑚)
64	3, 62, 63	sylancr 412	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁‘(◡𝑁‘𝑚)) = 𝑚)
65	61, 64	breq12d 4002	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((𝑁‘(◡𝑁‘𝑖)) ≤ (𝑁‘(◡𝑁‘𝑚)) ↔ 𝑖 ≤ 𝑚))
66	59, 65	bitrd 187	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((◡𝑁‘𝑖) ⊆ (◡𝑁‘𝑚) ↔ 𝑖 ≤ 𝑚))
67	54, 66	mpbird 166	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (◡𝑁‘𝑖) ⊆ (◡𝑁‘𝑚))
68		nnsssuc 6481	. . . . . . . . . . . . . 14 ⊢ (((◡𝑁‘𝑖) ∈ ω ∧ (◡𝑁‘𝑚) ∈ ω) → ((◡𝑁‘𝑖) ⊆ (◡𝑁‘𝑚) ↔ (◡𝑁‘𝑖) ∈ suc (◡𝑁‘𝑚)))
69	57, 58, 68	syl2anc 409	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((◡𝑁‘𝑖) ⊆ (◡𝑁‘𝑚) ↔ (◡𝑁‘𝑖) ∈ suc (◡𝑁‘𝑚)))
70	67, 69	mpbid 146	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (◡𝑁‘𝑖) ∈ suc (◡𝑁‘𝑚))
71	1	ad3antrrr 489	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝐹:ω–onto→𝐴)
72		fof 5420	. . . . . . . . . . . . . . 15 ⊢ (𝐹:ω–onto→𝐴 → 𝐹:ω⟶𝐴)
73	71, 72	syl 14	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝐹:ω⟶𝐴)
74	73	ffund 5351	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → Fun 𝐹)
75	73	fdmd 5354	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → dom 𝐹 = ω)
76	57, 75	eleqtrrd 2250	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (◡𝑁‘𝑖) ∈ dom 𝐹)
77		funfvima 5727	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ (◡𝑁‘𝑖) ∈ dom 𝐹) → ((◡𝑁‘𝑖) ∈ suc (◡𝑁‘𝑚) → (𝐹‘(◡𝑁‘𝑖)) ∈ (𝐹 “ suc (◡𝑁‘𝑚))))
78	74, 76, 77	syl2anc 409	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((◡𝑁‘𝑖) ∈ suc (◡𝑁‘𝑚) → (𝐹‘(◡𝑁‘𝑖)) ∈ (𝐹 “ suc (◡𝑁‘𝑚))))
79	70, 78	mpd 13	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐹‘(◡𝑁‘𝑖)) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))
80	52, 79	eqeltrd 2247	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘𝑖) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))
81	80	adantr 274	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) ∧ (𝐺‘(𝑁‘𝑘)) = (𝐺‘𝑖)) → (𝐺‘𝑖) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))
82	46, 81	eqeltrd 2247	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) ∧ (𝐺‘(𝑁‘𝑘)) = (𝐺‘𝑖)) → (𝐺‘(𝑁‘𝑘)) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))
83	45, 82	mtand 660	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ¬ (𝐺‘(𝑁‘𝑘)) = (𝐺‘𝑖))
84	83	neqned 2347	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖))
85	84	ralrimiva 2543	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) → ∀𝑖 ∈ (0...𝑚)(𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖))
86		fveq2 5496	. . . . . . . 8 ⊢ (𝑗 = (𝑁‘𝑘) → (𝐺‘𝑗) = (𝐺‘(𝑁‘𝑘)))
87	86	neeq1d 2358	. . . . . . 7 ⊢ (𝑗 = (𝑁‘𝑘) → ((𝐺‘𝑗) ≠ (𝐺‘𝑖) ↔ (𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖)))
88	87	ralbidv 2470	. . . . . 6 ⊢ (𝑗 = (𝑁‘𝑘) → (∀𝑖 ∈ (0...𝑚)(𝐺‘𝑗) ≠ (𝐺‘𝑖) ↔ ∀𝑖 ∈ (0...𝑚)(𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖)))
89	88	rspcev 2834	. . . . 5 ⊢ (((𝑁‘𝑘) ∈ ℕ0 ∧ ∀𝑖 ∈ (0...𝑚)(𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖)) → ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝐺‘𝑗) ≠ (𝐺‘𝑖))
90	32, 85, 89	syl2anc 409	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) → ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝐺‘𝑗) ≠ (𝐺‘𝑖))
91	27, 90	rexlimddv 2592	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝐺‘𝑗) ≠ (𝐺‘𝑖))
92	91	ralrimiva 2543	. 2 ⊢ (𝜑 → ∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝐺‘𝑗) ≠ (𝐺‘𝑖))
93	13, 92	jca 304	1 ⊢ (𝜑 → (𝐺:ℕ0–onto→𝐴 ∧ ∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝐺‘𝑗) ≠ (𝐺‘𝑖)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141   ≠ wne 2340  ∀wral 2448  ∃wrex 2449   ⊆ wss 3121   class class class wbr 3989   ↦ cmpt 4050  suc csuc 4350  ωcom 4574  ^◡ccnv 4610  dom cdm 4611   “ cima 4614   ∘ ccom 4615  Fun wfun 5192  ⟶wf 5194  –onto→wfo 5196  –1-1-onto→wf1o 5197  ‘cfv 5198  (class class class)co 5853  freccfrec 6369  0cc0 7774  1c1 7775   + caddc 7777   ≤ cle 7955  ℕ₀cn0 9135  ℤcz 9212  ...cfz 9965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966

This theorem is referenced by:  ctinfom  12383