Theorem ctinfomlemom 13178

Description: Lemma for ctinfom 13179. 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 10790	. . . . . 6 ⊢ 𝑁:ω–1-1-onto→ℕ0
4		f1ocnv 5627	. . . . . 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 5621	. . . . 5 ⊢ (◡𝑁:ℕ0–1-1-onto→ω → ◡𝑁:ℕ0–onto→ω)
7	5, 6	ax-mp 5	. . . 4 ⊢ ◡𝑁:ℕ0–onto→ω
8		foco 5601	. . . 4 ⊢ ((𝐹:ω–onto→𝐴 ∧ ◡𝑁:ℕ0–onto→ω) → (𝐹 ∘ ◡𝑁):ℕ0–onto→𝐴)
9	1, 7, 8	sylancl 413	. . 3 ⊢ (𝜑 → (𝐹 ∘ ◡𝑁):ℕ0–onto→𝐴)
10		ctinfom.g	. . . 4 ⊢ 𝐺 = (𝐹 ∘ ◡𝑁)
11		foeq1 5586	. . . 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 134	. 2 ⊢ (𝜑 → 𝐺:ℕ0–onto→𝐴)
14		imaeq2 5097	. . . . . . . 8 ⊢ (𝑛 = suc (◡𝑁‘𝑚) → (𝐹 “ 𝑛) = (𝐹 “ suc (◡𝑁‘𝑚)))
15	14	eleq2d 2302	. . . . . . 7 ⊢ (𝑛 = suc (◡𝑁‘𝑚) → ((𝐹‘𝑘) ∈ (𝐹 “ 𝑛) ↔ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚))))
16	15	notbid 673	. . . . . 6 ⊢ (𝑛 = suc (◡𝑁‘𝑚) → (¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑛) ↔ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚))))
17	16	rexbidv 2543	. . . . 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 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑛))
20		f1of 5614	. . . . . . . . 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 110	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
24	22, 23	ffvelcdmd 5813	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (◡𝑁‘𝑚) ∈ ω)
25		peano2 4717	. . . . . 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 2926	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ∃𝑘 ∈ ω ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))
28		f1of 5614	. . . . . . . 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 531	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) → 𝑘 ∈ ω)
32	30, 31	ffvelcdmd 5813	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) → (𝑁‘𝑘) ∈ ℕ0)
33	10	fveq1i 5671	. . . . . . . . . . 11 ⊢ (𝐺‘(𝑁‘𝑘)) = ((𝐹 ∘ ◡𝑁)‘(𝑁‘𝑘))
34	32	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁‘𝑘) ∈ ℕ0)
35		fvco3 5748	. . . . . . . . . . . 12 ⊢ ((◡𝑁:ℕ0⟶ω ∧ (𝑁‘𝑘) ∈ ℕ0) → ((𝐹 ∘ ◡𝑁)‘(𝑁‘𝑘)) = (𝐹‘(◡𝑁‘(𝑁‘𝑘))))
36	21, 34, 35	sylancr 414	. . . . . . . . . . 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 2277	. . . . . . . . . 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 276	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑘 ∈ ω)
39		f1ocnvfv1 5950	. . . . . . . . . . . . 13 ⊢ ((𝑁:ω–1-1-onto→ℕ0 ∧ 𝑘 ∈ ω) → (◡𝑁‘(𝑁‘𝑘)) = 𝑘)
40	3, 39	mpan 424	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ω → (◡𝑁‘(𝑁‘𝑘)) = 𝑘)
41	40	fveq2d 5674	. . . . . . . . . . 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 2265	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘(𝑁‘𝑘)) = (𝐹‘𝑘))
44		simplrr 538	. . . . . . . . 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 2328	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ¬ (𝐺‘(𝑁‘𝑘)) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))
46		simpr 110	. . . . . . . . 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 5671	. . . . . . . . . . . 12 ⊢ (𝐺‘𝑖) = ((𝐹 ∘ ◡𝑁)‘𝑖)
48		elfznn0 10448	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0...𝑚) → 𝑖 ∈ ℕ0)
49	48	adantl 277	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑖 ∈ ℕ0)
50		fvco3 5748	. . . . . . . . . . . . 13 ⊢ ((◡𝑁:ℕ0⟶ω ∧ 𝑖 ∈ ℕ0) → ((𝐹 ∘ ◡𝑁)‘𝑖) = (𝐹‘(◡𝑁‘𝑖)))
51	21, 49, 50	sylancr 414	. . . . . . . . . . . 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 2277	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘𝑖) = (𝐹‘(◡𝑁‘𝑖)))
53		elfzle2 10362	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0...𝑚) → 𝑖 ≤ 𝑚)
54	53	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑖 ≤ 𝑚)
55		0zd 9589	. . . . . . . . . . . . . . . 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	ffvelcdmd 5813	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (◡𝑁‘𝑖) ∈ ω)
58	24	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (◡𝑁‘𝑚) ∈ ω)
59	55, 2, 57, 58	frec2uzled 10791	. . . . . . . . . . . . . . 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 5951	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁:ω–1-1-onto→ℕ0 ∧ 𝑖 ∈ ℕ0) → (𝑁‘(◡𝑁‘𝑖)) = 𝑖)
61	3, 49, 60	sylancr 414	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁‘(◡𝑁‘𝑖)) = 𝑖)
62	23	ad2antrr 488	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑚 ∈ ℕ0)
63		f1ocnvfv2 5951	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁:ω–1-1-onto→ℕ0 ∧ 𝑚 ∈ ℕ0) → (𝑁‘(◡𝑁‘𝑚)) = 𝑚)
64	3, 62, 63	sylancr 414	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁‘(◡𝑁‘𝑚)) = 𝑚)
65	61, 64	breq12d 4122	. . . . . . . . . . . . . . 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 188	. . . . . . . . . . . . . 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 167	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (◡𝑁‘𝑖) ⊆ (◡𝑁‘𝑚))
68		nnsssuc 6735	. . . . . . . . . . . . . 14 ⊢ (((◡𝑁‘𝑖) ∈ ω ∧ (◡𝑁‘𝑚) ∈ ω) → ((◡𝑁‘𝑖) ⊆ (◡𝑁‘𝑚) ↔ (◡𝑁‘𝑖) ∈ suc (◡𝑁‘𝑚)))
69	57, 58, 68	syl2anc 411	. . . . . . . . . . . . 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 147	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (◡𝑁‘𝑖) ∈ suc (◡𝑁‘𝑚))
71	1	ad3antrrr 492	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝐹:ω–onto→𝐴)
72		fof 5590	. . . . . . . . . . . . . . 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 5512	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → Fun 𝐹)
75	73	fdmd 5515	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → dom 𝐹 = ω)
76	57, 75	eleqtrrd 2312	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (◡𝑁‘𝑖) ∈ dom 𝐹)
77		funfvima 5918	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ (◡𝑁‘𝑖) ∈ dom 𝐹) → ((◡𝑁‘𝑖) ∈ suc (◡𝑁‘𝑚) → (𝐹‘(◡𝑁‘𝑖)) ∈ (𝐹 “ suc (◡𝑁‘𝑚))))
78	74, 76, 77	syl2anc 411	. . . . . . . . . . . 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 2309	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘𝑖) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))
81	80	adantr 276	. . . . . . . . 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 2309	. . . . . . . 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 671	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ¬ (𝐺‘(𝑁‘𝑘)) = (𝐺‘𝑖))
84	83	neqned 2419	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖))
85	84	ralrimiva 2615	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (◡𝑁‘𝑚)))) → ∀𝑖 ∈ (0...𝑚)(𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖))
86		fveq2 5670	. . . . . . . 8 ⊢ (𝑗 = (𝑁‘𝑘) → (𝐺‘𝑗) = (𝐺‘(𝑁‘𝑘)))
87	86	neeq1d 2430	. . . . . . 7 ⊢ (𝑗 = (𝑁‘𝑘) → ((𝐺‘𝑗) ≠ (𝐺‘𝑖) ↔ (𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖)))
88	87	ralbidv 2542	. . . . . 6 ⊢ (𝑗 = (𝑁‘𝑘) → (∀𝑖 ∈ (0...𝑚)(𝐺‘𝑗) ≠ (𝐺‘𝑖) ↔ ∀𝑖 ∈ (0...𝑚)(𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖)))
89	88	rspcev 2921	. . . . 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 411	. . . 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 2665	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝐺‘𝑗) ≠ (𝐺‘𝑖))
92	91	ralrimiva 2615	. 2 ⊢ (𝜑 → ∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝐺‘𝑗) ≠ (𝐺‘𝑖))
93	13, 92	jca 306	1 ⊢ (𝜑 → (𝐺:ℕ0–onto→𝐴 ∧ ∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝐺‘𝑗) ≠ (𝐺‘𝑖)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203   ≠ wne 2412  ∀wral 2520  ∃wrex 2521   ⊆ wss 3211   class class class wbr 4109   ↦ cmpt 4171  suc csuc 4486  ωcom 4712  ^◡ccnv 4748  dom cdm 4749   “ cima 4752   ∘ ccom 4753  Fun wfun 5346  ⟶wf 5348  –onto→wfo 5350  –1-1-onto→wf1o 5351  ‘cfv 5352  (class class class)co 6050  freccfrec 6621  0cc0 8127  1c1 8128   + caddc 8130   ≤ cle 8309  ℕ₀cn0 9496  ℤcz 9577  ...cfz 10342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343

This theorem is referenced by:  ctinfom  13179