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Theorem ctinfomlemom 12453

Description: Lemma for ctinfom 12454. 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 10448	. . . . . 6 ⊢ 𝑁:ω–1-1-onto→ℕ₀
4		f1ocnv 5490	. . . . . 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 5484	. . . . 5 ⊢ (^◡𝑁:ℕ₀–1-1-onto→ω → ^◡𝑁:ℕ₀–onto→ω)
7	5, 6	ax-mp 5	. . . 4 ⊢ ^◡𝑁:ℕ₀–onto→ω
8		foco 5464	. . . 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 5450	. . . 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 4981	. . . . . . . 8 ⊢ (𝑛 = suc (^◡𝑁‘𝑚) → (𝐹 “ 𝑛) = (𝐹 “ suc (^◡𝑁‘𝑚)))
15	14	eleq2d 2259	. . . . . . 7 ⊢ (𝑛 = suc (^◡𝑁‘𝑚) → ((𝐹‘𝑘) ∈ (𝐹 “ 𝑛) ↔ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚))))
16	15	notbid 668	. . . . . 6 ⊢ (𝑛 = suc (^◡𝑁‘𝑚) → (¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑛) ↔ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚))))
17	16	rexbidv 2491	. . . . 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 5477	. . . . . . . . 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 5669	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (^◡𝑁‘𝑚) ∈ ω)
25		peano2 4609	. . . . . 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 2861	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ∃𝑘 ∈ ω ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))
28		f1of 5477	. . . . . . . 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 5669	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) → (𝑁‘𝑘) ∈ ℕ₀)
33	10	fveq1i 5532	. . . . . . . . . . 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 5604	. . . . . . . . . . . 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 2234	. . . . . . . . . 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 5795	. . . . . . . . . . . . 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 5535	. . . . . . . . . . 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 2222	. . . . . . . . 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 2285	. . . . . . . 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 5532	. . . . . . . . . . . 12 ⊢ (𝐺‘𝑖) = ((𝐹 ∘ ^◡𝑁)‘𝑖)
48		elfznn0 10134	. . . . . . . . . . . . . 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 5604	. . . . . . . . . . . . 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 2234	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘𝑖) = (𝐹‘(^◡𝑁‘𝑖)))
53		elfzle2 10048	. . . . . . . . . . . . . . 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 9285	. . . . . . . . . . . . . . . 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 5669	. . . . . . . . . . . . . . . 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 10449	. . . . . . . . . . . . . . 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 5796	. . . . . . . . . . . . . . . . 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 5796	. . . . . . . . . . . . . . . . 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 4031	. . . . . . . . . . . . . . 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 6522	. . . . . . . . . . . . . 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 5454	. . . . . . . . . . . . . . 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 5385	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → Fun 𝐹)
75	73	fdmd 5388	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → dom 𝐹 = ω)
76	57, 75	eleqtrrd 2269	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (^◡𝑁‘𝑖) ∈ dom 𝐹)
77		funfvima 5765	. . . . . . . . . . . . 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 2266	. . . . . . . . . 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 2266	. . . . . . . 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 666	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ¬ (𝐺‘(𝑁‘𝑘)) = (𝐺‘𝑖))
84	83	neqned 2367	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖))
85	84	ralrimiva 2563	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ suc (^◡𝑁‘𝑚)))) → ∀𝑖 ∈ (0...𝑚)(𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖))
86		fveq2 5531	. . . . . . . 8 ⊢ (𝑗 = (𝑁‘𝑘) → (𝐺‘𝑗) = (𝐺‘(𝑁‘𝑘)))
87	86	neeq1d 2378	. . . . . . 7 ⊢ (𝑗 = (𝑁‘𝑘) → ((𝐺‘𝑗) ≠ (𝐺‘𝑖) ↔ (𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖)))
88	87	ralbidv 2490	. . . . . 6 ⊢ (𝑗 = (𝑁‘𝑘) → (∀𝑖 ∈ (0...𝑚)(𝐺‘𝑗) ≠ (𝐺‘𝑖) ↔ ∀𝑖 ∈ (0...𝑚)(𝐺‘(𝑁‘𝑘)) ≠ (𝐺‘𝑖)))
89	88	rspcev 2856	. . . . 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 2612	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ∃𝑗 ∈ ℕ₀ ∀𝑖 ∈ (0...𝑚)(𝐺‘𝑗) ≠ (𝐺‘𝑖))
92	91	ralrimiva 2563	. 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 1364 ∈ wcel 2160 ≠ wne 2360 ∀wral 2468 ∃wrex 2469 ⊆ wss 3144 class class class wbr 4018 ↦ cmpt 4079 suc csuc 4380 ωcom 4604 ^◡ccnv 4640 dom cdm 4641 “ cima 4644 ∘ ccom 4645 Fun wfun 5226 ⟶wf 5228 –onto→wfo 5230 –1-1-onto→wf1o 5231 ‘cfv 5232 (class class class)co 5892 freccfrec 6410 0cc0 7831 1c1 7832 + caddc 7834 ≤ cle 8013 ℕ₀cn0 9196 ℤcz 9273 ...cfz 10028

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7922 ax-resscn 7923 ax-1cn 7924 ax-1re 7925 ax-icn 7926 ax-addcl 7927 ax-addrcl 7928 ax-mulcl 7929 ax-addcom 7931 ax-addass 7933 ax-distr 7935 ax-i2m1 7936 ax-0lt1 7937 ax-0id 7939 ax-rnegex 7940 ax-cnre 7942 ax-pre-ltirr 7943 ax-pre-ltwlin 7944 ax-pre-lttrn 7945 ax-pre-ltadd 7947

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-recs 6325 df-frec 6411 df-pnf 8014 df-mnf 8015 df-xr 8016 df-ltxr 8017 df-le 8018 df-sub 8150 df-neg 8151 df-inn 8940 df-n0 9197 df-z 9274 df-uz 9549 df-fz 10029

This theorem is referenced by: ctinfom 12454

W3C validator