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Theorem ctinfomlemom 12430

Description: Lemma for ctinfom 12431. 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 10430	. . . . . 6 ⊢ 𝑁:ω–1-1-onto→ℕ₀
4		f1ocnv 5476	. . . . . 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 5470	. . . . 5 ⊢ (^◡𝑁:ℕ₀–1-1-onto→ω → ^◡𝑁:ℕ₀–onto→ω)
7	5, 6	ax-mp 5	. . . 4 ⊢ ^◡𝑁:ℕ₀–onto→ω
8		foco 5450	. . . 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 5436	. . . 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 4968	. . . . . . . 8 ⊢ (𝑛 = suc (^◡𝑁‘𝑚) → (𝐹 “ 𝑛) = (𝐹 “ suc (^◡𝑁‘𝑚)))
15	14	eleq2d 2247	. . . . . . 7 ⊢ (𝑛 = suc (^◡𝑁‘𝑚) → ((𝐹‘𝑘) ∈ (𝐹 “ 𝑛) ↔ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚))))
16	15	notbid 667	. . . . . 6 ⊢ (𝑛 = suc (^◡𝑁‘𝑚) → (¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑛) ↔ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚))))
17	16	rexbidv 2478	. . . . 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 5463	. . . . . . . . 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 5654	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (^◡𝑁‘𝑚) ∈ ω)
25		peano2 4596	. . . . . 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 2848	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ∃𝑘 ∈ ω ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))
28		f1of 5463	. . . . . . . 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 529	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) → 𝑘 ∈ ω)
32	30, 31	ffvelcdmd 5654	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) → (𝑁‘𝑘) ∈ ℕ₀)
33	10	fveq1i 5518	. . . . . . . . . . 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 5589	. . . . . . . . . . . 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 2222	. . . . . . . . . 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 5780	. . . . . . . . . . . . 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 5521	. . . . . . . . . . 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 2210	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘(𝑁‘𝑘)) = (𝐹‘𝑘))
44		simplrr 536	. . . . . . . . 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 2273	. . . . . . . 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 5518	. . . . . . . . . . . 12 ⊢ (𝐺‘𝑖) = ((𝐹 ∘ ^◡𝑁)‘𝑖)
48		elfznn0 10116	. . . . . . . . . . . . . 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 5589	. . . . . . . . . . . . 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 2222	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘𝑖) = (𝐹‘(^◡𝑁‘𝑖)))
53		elfzle2 10030	. . . . . . . . . . . . . . 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 9267	. . . . . . . . . . . . . . . 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 5654	. . . . . . . . . . . . . . . 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 10431	. . . . . . . . . . . . . . 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 5781	. . . . . . . . . . . . . . . . 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 5781	. . . . . . . . . . . . . . . . 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 4018	. . . . . . . . . . . . . . 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 6505	. . . . . . . . . . . . . 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 5440	. . . . . . . . . . . . . . 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 5371	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → Fun 𝐹)
75	73	fdmd 5374	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → dom 𝐹 = ω)
76	57, 75	eleqtrrd 2257	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (^◡𝑁‘𝑖) ∈ dom 𝐹)
77		funfvima 5750	. . . . . . . . . . . . 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 2254	. . . . . . . . . 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 2254	. . . . . . . 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 665	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ¬ (𝐺‘(𝑁‘𝑘)) = (𝐺‘𝑖))
84	83	neqned 2354	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖))
85	84	ralrimiva 2550	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) → ∀𝑖 ∈ (0...𝑚)(𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖))
86		fveq2 5517	. . . . . . . 8 ⊢ (𝑗 = (𝑁‘𝑘) → (𝐺‘𝑗) = (𝐺‘(𝑁‘𝑘)))
87	86	neeq1d 2365	. . . . . . 7 ⊢ (𝑗 = (𝑁‘𝑘) → ((𝐺‘𝑗) ≠ (𝐺‘𝑖) ↔ (𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖)))
88	87	ralbidv 2477	. . . . . 6 ⊢ (𝑗 = (𝑁‘𝑘) → (∀𝑖 ∈ (0...𝑚)(𝐺‘𝑗) ≠ (𝐺‘𝑖) ↔ ∀𝑖 ∈ (0...𝑚)(𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖)))
89	88	rspcev 2843	. . . . 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 2599	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ∃𝑗 ∈ ℕ₀ ∀𝑖 ∈ (0...𝑚)(𝐺‘𝑗) ≠ (𝐺‘𝑖))
92	91	ralrimiva 2550	. 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)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∀wral 2455 ∃wrex 2456 ⊆ wss 3131 class class class wbr 4005 ↦ cmpt 4066 suc csuc 4367 ωcom 4591 ^◡ccnv 4627 dom cdm 4628 “ cima 4631 ∘ ccom 4632 Fun wfun 5212 ⟶wf 5214 –onto→wfo 5216 –1-1-onto→wf1o 5217 ‘cfv 5218 (class class class)co 5877 freccfrec 6393 0cc0 7813 1c1 7814 + caddc 7816 ≤ cle 7995 ℕ₀cn0 9178 ℤcz 9255 ...cfz 10010

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-addass 7915 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-0id 7921 ax-rnegex 7922 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-ltadd 7929

This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-recs 6308 df-frec 6394 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-inn 8922 df-n0 9179 df-z 9256 df-uz 9531 df-fz 10011

This theorem is referenced by: ctinfom 12431

W3C validator