Theorem ennnfonelemnn0 13291

Description: Lemma for ennnfone 13294. A version of ennnfonelemen 13290 expressed in terms of ℕ₀ instead of ω. (Contributed by Jim Kingdon, 27-Oct-2022.)

Hypotheses

Ref	Expression
ennnfonelemr.dceq	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
ennnfonelemr.f	⊢ (𝜑 → 𝐹:ℕ0–onto→𝐴)
ennnfonelemr.n	⊢ (𝜑 → ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗))
ennnfonelemnn0.n	⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)

Assertion

Ref	Expression
ennnfonelemnn0	⊢ (𝜑 → 𝐴 ≈ ℕ)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐹,𝑦   𝑗,𝐹,𝑘,𝑛   𝑥,𝑁,𝑦   𝑗,𝑁,𝑘,𝑛   𝜑,𝑘   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑗,𝑛)   𝐴(𝑗,𝑘,𝑛)

Proof of Theorem ennnfonelemnn0

Dummy variables 𝑎 𝑏 𝑐 𝑖 𝑟 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ennnfonelemr.dceq	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
2		ennnfonelemr.f	. . 3 ⊢ (𝜑 → 𝐹:ℕ0–onto→𝐴)
3		ennnfonelemnn0.n	. . . . . 6 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
4	3	frechashgf1o 10843	. . . . 5 ⊢ 𝑁:ω–1-1-onto→ℕ0
5		f1ofo 5641	. . . . 5 ⊢ (𝑁:ω–1-1-onto→ℕ0 → 𝑁:ω–onto→ℕ0)
6	4, 5	ax-mp 5	. . . 4 ⊢ 𝑁:ω–onto→ℕ0
7	6	a1i 9	. . 3 ⊢ (𝜑 → 𝑁:ω–onto→ℕ0)
8		foco 5621	. . 3 ⊢ ((𝐹:ℕ0–onto→𝐴 ∧ 𝑁:ω–onto→ℕ0) → (𝐹 ∘ 𝑁):ω–onto→𝐴)
9	2, 7, 8	syl2anc 415	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝑁):ω–onto→𝐴)
10		oveq2 6083	. . . . . . 7 ⊢ (𝑛 = (𝑁‘𝑝) → (0...𝑛) = (0...(𝑁‘𝑝)))
11	10	raleqdv 2755	. . . . . 6 ⊢ (𝑛 = (𝑁‘𝑝) → (∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗)))
12	11	rexbidv 2551	. . . . 5 ⊢ (𝑛 = (𝑁‘𝑝) → (∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗)))
13		ennnfonelemr.n	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗))
14	13	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ ω) → ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗))
15		f1of 5634	. . . . . . . 8 ⊢ (𝑁:ω–1-1-onto→ℕ0 → 𝑁:ω⟶ℕ0)
16	4, 15	ax-mp 5	. . . . . . 7 ⊢ 𝑁:ω⟶ℕ0
17	16	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ω) → 𝑁:ω⟶ℕ0)
18		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ω) → 𝑝 ∈ ω)
19	17, 18	ffvelcdmd 5835	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ ω) → (𝑁‘𝑝) ∈ ℕ0)
20	12, 14, 19	rspcdva 2934	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ ω) → ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))
21		f1ocnv 5647	. . . . . . . 8 ⊢ (𝑁:ω–1-1-onto→ℕ0 → ◡𝑁:ℕ0–1-1-onto→ω)
22		f1of 5634	. . . . . . . 8 ⊢ (◡𝑁:ℕ0–1-1-onto→ω → ◡𝑁:ℕ0⟶ω)
23	4, 21, 22	mp2b 8	. . . . . . 7 ⊢ ◡𝑁:ℕ0⟶ω
24	23	a1i 9	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ◡𝑁:ℕ0⟶ω)
25		simprl 535	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → 𝑘 ∈ ℕ0)
26	24, 25	ffvelcdmd 5835	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (◡𝑁‘𝑘) ∈ ω)
27		fveq2 5690	. . . . . . . . 9 ⊢ (𝑗 = (𝑁‘𝑟) → (𝐹‘𝑗) = (𝐹‘(𝑁‘𝑟)))
28	27	neeq2d 2439	. . . . . . . 8 ⊢ (𝑗 = (𝑁‘𝑟) → ((𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ (𝐹‘𝑘) ≠ (𝐹‘(𝑁‘𝑟))))
29		simplrr 542	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))
30		simpr 110	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑟 ∈ suc 𝑝)
31	18	ad2antrr 492	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑝 ∈ ω)
32		peano2 4737	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ω → suc 𝑝 ∈ ω)
33	31, 32	syl 14	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → suc 𝑝 ∈ ω)
34		elnn 4748	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ suc 𝑝 ∧ suc 𝑝 ∈ ω) → 𝑟 ∈ ω)
35	30, 33, 34	syl2anc 415	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑟 ∈ ω)
36	16	ffvelcdmi 5833	. . . . . . . . . 10 ⊢ (𝑟 ∈ ω → (𝑁‘𝑟) ∈ ℕ0)
37	35, 36	syl 14	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘𝑟) ∈ ℕ0)
38		0zd 9635	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 0 ∈ ℤ)
39	38, 3, 35, 33	frec2uzltd 10818	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑟 ∈ suc 𝑝 → (𝑁‘𝑟) < (𝑁‘suc 𝑝)))
40	30, 39	mpd 13	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘𝑟) < (𝑁‘suc 𝑝))
41	38, 3, 31	frec2uzsucd 10816	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘suc 𝑝) = ((𝑁‘𝑝) + 1))
42	40, 41	breqtrd 4151	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘𝑟) < ((𝑁‘𝑝) + 1))
43	19	ad2antrr 492	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘𝑝) ∈ ℕ0)
44		nn0leltp1 9687	. . . . . . . . . . 11 ⊢ (((𝑁‘𝑟) ∈ ℕ0 ∧ (𝑁‘𝑝) ∈ ℕ0) → ((𝑁‘𝑟) ≤ (𝑁‘𝑝) ↔ (𝑁‘𝑟) < ((𝑁‘𝑝) + 1)))
45	37, 43, 44	syl2anc 415	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝑁‘𝑟) ≤ (𝑁‘𝑝) ↔ (𝑁‘𝑟) < ((𝑁‘𝑝) + 1)))
46	42, 45	mpbird 167	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘𝑟) ≤ (𝑁‘𝑝))
47		fznn0 10498	. . . . . . . . . 10 ⊢ ((𝑁‘𝑝) ∈ ℕ0 → ((𝑁‘𝑟) ∈ (0...(𝑁‘𝑝)) ↔ ((𝑁‘𝑟) ∈ ℕ0 ∧ (𝑁‘𝑟) ≤ (𝑁‘𝑝))))
48	43, 47	syl 14	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝑁‘𝑟) ∈ (0...(𝑁‘𝑝)) ↔ ((𝑁‘𝑟) ∈ ℕ0 ∧ (𝑁‘𝑟) ≤ (𝑁‘𝑝))))
49	37, 46, 48	mpbir2and 957	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘𝑟) ∈ (0...(𝑁‘𝑝)))
50	28, 29, 49	rspcdva 2934	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝐹‘𝑘) ≠ (𝐹‘(𝑁‘𝑟)))
51	26	adantr 276	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (◡𝑁‘𝑘) ∈ ω)
52		fvco3 5770	. . . . . . . . 9 ⊢ ((𝑁:ω⟶ℕ0 ∧ (◡𝑁‘𝑘) ∈ ω) → ((𝐹 ∘ 𝑁)‘(◡𝑁‘𝑘)) = (𝐹‘(𝑁‘(◡𝑁‘𝑘))))
53	16, 51, 52	sylancr 418	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹 ∘ 𝑁)‘(◡𝑁‘𝑘)) = (𝐹‘(𝑁‘(◡𝑁‘𝑘))))
54	25	adantr 276	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑘 ∈ ℕ0)
55		f1ocnvfv2 5974	. . . . . . . . . 10 ⊢ ((𝑁:ω–1-1-onto→ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑁‘(◡𝑁‘𝑘)) = 𝑘)
56	4, 54, 55	sylancr 418	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘(◡𝑁‘𝑘)) = 𝑘)
57	56	fveq2d 5694	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝐹‘(𝑁‘(◡𝑁‘𝑘))) = (𝐹‘𝑘))
58	53, 57	eqtrd 2271	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹 ∘ 𝑁)‘(◡𝑁‘𝑘)) = (𝐹‘𝑘))
59		fvco3 5770	. . . . . . . 8 ⊢ ((𝑁:ω⟶ℕ0 ∧ 𝑟 ∈ ω) → ((𝐹 ∘ 𝑁)‘𝑟) = (𝐹‘(𝑁‘𝑟)))
60	16, 35, 59	sylancr 418	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹 ∘ 𝑁)‘𝑟) = (𝐹‘(𝑁‘𝑟)))
61	50, 58, 60	3netr4d 2453	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹 ∘ 𝑁)‘(◡𝑁‘𝑘)) ≠ ((𝐹 ∘ 𝑁)‘𝑟))
62	61	ralrimiva 2623	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ∀𝑟 ∈ suc 𝑝((𝐹 ∘ 𝑁)‘(◡𝑁‘𝑘)) ≠ ((𝐹 ∘ 𝑁)‘𝑟))
63		fveq2 5690	. . . . . . . 8 ⊢ (𝑞 = (◡𝑁‘𝑘) → ((𝐹 ∘ 𝑁)‘𝑞) = ((𝐹 ∘ 𝑁)‘(◡𝑁‘𝑘)))
64	63	neeq1d 2438	. . . . . . 7 ⊢ (𝑞 = (◡𝑁‘𝑘) → (((𝐹 ∘ 𝑁)‘𝑞) ≠ ((𝐹 ∘ 𝑁)‘𝑟) ↔ ((𝐹 ∘ 𝑁)‘(◡𝑁‘𝑘)) ≠ ((𝐹 ∘ 𝑁)‘𝑟)))
65	64	ralbidv 2550	. . . . . 6 ⊢ (𝑞 = (◡𝑁‘𝑘) → (∀𝑟 ∈ suc 𝑝((𝐹 ∘ 𝑁)‘𝑞) ≠ ((𝐹 ∘ 𝑁)‘𝑟) ↔ ∀𝑟 ∈ suc 𝑝((𝐹 ∘ 𝑁)‘(◡𝑁‘𝑘)) ≠ ((𝐹 ∘ 𝑁)‘𝑟)))
66	65	rspcev 2929	. . . . 5 ⊢ (((◡𝑁‘𝑘) ∈ ω ∧ ∀𝑟 ∈ suc 𝑝((𝐹 ∘ 𝑁)‘(◡𝑁‘𝑘)) ≠ ((𝐹 ∘ 𝑁)‘𝑟)) → ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹 ∘ 𝑁)‘𝑞) ≠ ((𝐹 ∘ 𝑁)‘𝑟))
67	26, 62, 66	syl2anc 415	. . . 4 ⊢ (((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹 ∘ 𝑁)‘𝑞) ≠ ((𝐹 ∘ 𝑁)‘𝑟))
68	20, 67	rexlimddv 2673	. . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ ω) → ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹 ∘ 𝑁)‘𝑞) ≠ ((𝐹 ∘ 𝑁)‘𝑟))
69	68	ralrimiva 2623	. 2 ⊢ (𝜑 → ∀𝑝 ∈ ω ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹 ∘ 𝑁)‘𝑞) ≠ ((𝐹 ∘ 𝑁)‘𝑟))
70		id 19	. . . 4 ⊢ (𝑎 = 𝑥 → 𝑎 = 𝑥)
71		dmeq 4976	. . . . . . 7 ⊢ (𝑎 = 𝑥 → dom 𝑎 = dom 𝑥)
72	71	opeq1d 3905	. . . . . 6 ⊢ (𝑎 = 𝑥 → ⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩ = ⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩)
73	72	sneqd 3718	. . . . 5 ⊢ (𝑎 = 𝑥 → {⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩} = {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩})
74	70, 73	uneq12d 3384	. . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 ∪ {⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩}) = (𝑥 ∪ {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩}))
75	70, 74	ifeq12d 3657	. . 3 ⊢ (𝑎 = 𝑥 → if(((𝐹 ∘ 𝑁)‘𝑏) ∈ ((𝐹 ∘ 𝑁) “ 𝑏), 𝑎, (𝑎 ∪ {⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩})) = if(((𝐹 ∘ 𝑁)‘𝑏) ∈ ((𝐹 ∘ 𝑁) “ 𝑏), 𝑥, (𝑥 ∪ {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩})))
76		fveq2 5690	. . . . 5 ⊢ (𝑏 = 𝑦 → ((𝐹 ∘ 𝑁)‘𝑏) = ((𝐹 ∘ 𝑁)‘𝑦))
77		imaeq2 5117	. . . . 5 ⊢ (𝑏 = 𝑦 → ((𝐹 ∘ 𝑁) “ 𝑏) = ((𝐹 ∘ 𝑁) “ 𝑦))
78	76, 77	eleq12d 2309	. . . 4 ⊢ (𝑏 = 𝑦 → (((𝐹 ∘ 𝑁)‘𝑏) ∈ ((𝐹 ∘ 𝑁) “ 𝑏) ↔ ((𝐹 ∘ 𝑁)‘𝑦) ∈ ((𝐹 ∘ 𝑁) “ 𝑦)))
79	76	opeq2d 3906	. . . . . 6 ⊢ (𝑏 = 𝑦 → ⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩ = ⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑦)⟩)
80	79	sneqd 3718	. . . . 5 ⊢ (𝑏 = 𝑦 → {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩} = {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑦)⟩})
81	80	uneq2d 3383	. . . 4 ⊢ (𝑏 = 𝑦 → (𝑥 ∪ {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩}) = (𝑥 ∪ {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑦)⟩}))
82	78, 81	ifbieq2d 3662	. . 3 ⊢ (𝑏 = 𝑦 → if(((𝐹 ∘ 𝑁)‘𝑏) ∈ ((𝐹 ∘ 𝑁) “ 𝑏), 𝑥, (𝑥 ∪ {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩})) = if(((𝐹 ∘ 𝑁)‘𝑦) ∈ ((𝐹 ∘ 𝑁) “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑦)⟩})))
83	75, 82	cbvmpov 6158	. 2 ⊢ (𝑎 ∈ (𝐴 ↑pm ω), 𝑏 ∈ ω ↦ if(((𝐹 ∘ 𝑁)‘𝑏) ∈ ((𝐹 ∘ 𝑁) “ 𝑏), 𝑎, (𝑎 ∪ {⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩}))) = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if(((𝐹 ∘ 𝑁)‘𝑦) ∈ ((𝐹 ∘ 𝑁) “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑦)⟩})))
84		eqeq1 2245	. . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 = 0 ↔ 𝑥 = 0))
85		fvoveq1 6098	. . . 4 ⊢ (𝑎 = 𝑥 → (◡𝑁‘(𝑎 − 1)) = (◡𝑁‘(𝑥 − 1)))
86	84, 85	ifbieq2d 3662	. . 3 ⊢ (𝑎 = 𝑥 → if(𝑎 = 0, ∅, (◡𝑁‘(𝑎 − 1))) = if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))
87	86	cbvmptv 4222	. 2 ⊢ (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (◡𝑁‘(𝑎 − 1)))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))
88		eqid 2238	. 2 ⊢ seq0((𝑎 ∈ (𝐴 ↑pm ω), 𝑏 ∈ ω ↦ if(((𝐹 ∘ 𝑁)‘𝑏) ∈ ((𝐹 ∘ 𝑁) “ 𝑏), 𝑎, (𝑎 ∪ {⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩}))), (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (◡𝑁‘(𝑎 − 1))))) = seq0((𝑎 ∈ (𝐴 ↑pm ω), 𝑏 ∈ ω ↦ if(((𝐹 ∘ 𝑁)‘𝑏) ∈ ((𝐹 ∘ 𝑁) “ 𝑏), 𝑎, (𝑎 ∪ {⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩}))), (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (◡𝑁‘(𝑎 − 1)))))
89		fveq2 5690	. . 3 ⊢ (𝑖 = 𝑐 → (seq0((𝑎 ∈ (𝐴 ↑pm ω), 𝑏 ∈ ω ↦ if(((𝐹 ∘ 𝑁)‘𝑏) ∈ ((𝐹 ∘ 𝑁) “ 𝑏), 𝑎, (𝑎 ∪ {⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩}))), (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (◡𝑁‘(𝑎 − 1)))))‘𝑖) = (seq0((𝑎 ∈ (𝐴 ↑pm ω), 𝑏 ∈ ω ↦ if(((𝐹 ∘ 𝑁)‘𝑏) ∈ ((𝐹 ∘ 𝑁) “ 𝑏), 𝑎, (𝑎 ∪ {⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩}))), (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (◡𝑁‘(𝑎 − 1)))))‘𝑐))
90	89	cbviunv 4046	. 2 ⊢ ∪ 𝑖 ∈ ℕ0 (seq0((𝑎 ∈ (𝐴 ↑pm ω), 𝑏 ∈ ω ↦ if(((𝐹 ∘ 𝑁)‘𝑏) ∈ ((𝐹 ∘ 𝑁) “ 𝑏), 𝑎, (𝑎 ∪ {⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩}))), (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (◡𝑁‘(𝑎 − 1)))))‘𝑖) = ∪ 𝑐 ∈ ℕ0 (seq0((𝑎 ∈ (𝐴 ↑pm ω), 𝑏 ∈ ω ↦ if(((𝐹 ∘ 𝑁)‘𝑏) ∈ ((𝐹 ∘ 𝑁) “ 𝑏), 𝑎, (𝑎 ∪ {⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩}))), (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (◡𝑁‘(𝑎 − 1)))))‘𝑐)
91	1, 9, 69, 83, 3, 87, 88, 90	ennnfonelemen 13290	1 ⊢ (𝜑 → 𝐴 ≈ ℕ)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 846   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528  ∃wrex 2529   ∪ cun 3218  ∅c0 3520  ifcif 3635  {csn 3705  ⟨cop 3708  ∪ ciun 4007   class class class wbr 4125   ↦ cmpt 4187  suc csuc 4505  ωcom 4732  ^◡ccnv 4768  dom cdm 4769   “ cima 4772   ∘ ccom 4773  ⟶wf 5368  –onto→wfo 5370  –1-1-onto→wf1o 5371  ‘cfv 5372  (class class class)co 6075   ∈ cmpo 6077  freccfrec 6651   ↑_pm cpm 6913   ≈ cen 7010  0cc0 8169  1c1 8170   + caddc 8172   < clt 8350   ≤ cle 8351   − cmin 8487  ℕcn 9283  ℕ₀cn0 9542  ℤcz 9623  ...cfz 10390  seqcseq 10862

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-er 6797  df-pm 6915  df-en 7013  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-seqfrec 10863

This theorem is referenced by:  ennnfonelemr  13292