Theorem ennnfonelemnn0 12639

Description: Lemma for ennnfone 12642. A version of ennnfonelemen 12638 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 10520	. . . . 5 ⊢ 𝑁:ω–1-1-onto→ℕ0
5		f1ofo 5511	. . . . 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 5491	. . 3 ⊢ ((𝐹:ℕ0–onto→𝐴 ∧ 𝑁:ω–onto→ℕ0) → (𝐹 ∘ 𝑁):ω–onto→𝐴)
9	2, 7, 8	syl2anc 411	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝑁):ω–onto→𝐴)
10		oveq2 5930	. . . . . . 7 ⊢ (𝑛 = (𝑁‘𝑝) → (0...𝑛) = (0...(𝑁‘𝑝)))
11	10	raleqdv 2699	. . . . . 6 ⊢ (𝑛 = (𝑁‘𝑝) → (∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗)))
12	11	rexbidv 2498	. . . . 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 5504	. . . . . . . 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 5698	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ ω) → (𝑁‘𝑝) ∈ ℕ0)
20	12, 14, 19	rspcdva 2873	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ ω) → ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))
21		f1ocnv 5517	. . . . . . . 8 ⊢ (𝑁:ω–1-1-onto→ℕ0 → ◡𝑁:ℕ0–1-1-onto→ω)
22		f1of 5504	. . . . . . . 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 529	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → 𝑘 ∈ ℕ0)
26	24, 25	ffvelcdmd 5698	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (◡𝑁‘𝑘) ∈ ω)
27		fveq2 5558	. . . . . . . . 9 ⊢ (𝑗 = (𝑁‘𝑟) → (𝐹‘𝑗) = (𝐹‘(𝑁‘𝑟)))
28	27	neeq2d 2386	. . . . . . . 8 ⊢ (𝑗 = (𝑁‘𝑟) → ((𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ (𝐹‘𝑘) ≠ (𝐹‘(𝑁‘𝑟))))
29		simplrr 536	. . . . . . . 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 488	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑝 ∈ ω)
32		peano2 4631	. . . . . . . . . . . 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 4642	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ suc 𝑝 ∧ suc 𝑝 ∈ ω) → 𝑟 ∈ ω)
35	30, 33, 34	syl2anc 411	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑟 ∈ ω)
36	16	ffvelcdmi 5696	. . . . . . . . . 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 9338	. . . . . . . . . . . . 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 10495	. . . . . . . . . . . 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 10493	. . . . . . . . . . 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 4059	. . . . . . . . . 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 488	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘𝑝) ∈ ℕ0)
44		nn0leltp1 9389	. . . . . . . . . . 11 ⊢ (((𝑁‘𝑟) ∈ ℕ0 ∧ (𝑁‘𝑝) ∈ ℕ0) → ((𝑁‘𝑟) ≤ (𝑁‘𝑝) ↔ (𝑁‘𝑟) < ((𝑁‘𝑝) + 1)))
45	37, 43, 44	syl2anc 411	. . . . . . . . . 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 10188	. . . . . . . . . 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 946	. . . . . . . 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 2873	. . . . . . 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 5632	. . . . . . . . 9 ⊢ ((𝑁:ω⟶ℕ0 ∧ (◡𝑁‘𝑘) ∈ ω) → ((𝐹 ∘ 𝑁)‘(◡𝑁‘𝑘)) = (𝐹‘(𝑁‘(◡𝑁‘𝑘))))
53	16, 51, 52	sylancr 414	. . . . . . . 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 5825	. . . . . . . . . 10 ⊢ ((𝑁:ω–1-1-onto→ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑁‘(◡𝑁‘𝑘)) = 𝑘)
56	4, 54, 55	sylancr 414	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘(◡𝑁‘𝑘)) = 𝑘)
57	56	fveq2d 5562	. . . . . . . 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 2229	. . . . . . 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 5632	. . . . . . . 8 ⊢ ((𝑁:ω⟶ℕ0 ∧ 𝑟 ∈ ω) → ((𝐹 ∘ 𝑁)‘𝑟) = (𝐹‘(𝑁‘𝑟)))
60	16, 35, 59	sylancr 414	. . . . . . 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 2400	. . . . . 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 2570	. . . . 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 5558	. . . . . . . 8 ⊢ (𝑞 = (◡𝑁‘𝑘) → ((𝐹 ∘ 𝑁)‘𝑞) = ((𝐹 ∘ 𝑁)‘(◡𝑁‘𝑘)))
64	63	neeq1d 2385	. . . . . . 7 ⊢ (𝑞 = (◡𝑁‘𝑘) → (((𝐹 ∘ 𝑁)‘𝑞) ≠ ((𝐹 ∘ 𝑁)‘𝑟) ↔ ((𝐹 ∘ 𝑁)‘(◡𝑁‘𝑘)) ≠ ((𝐹 ∘ 𝑁)‘𝑟)))
65	64	ralbidv 2497	. . . . . 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 2868	. . . . 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 411	. . . 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 2619	. . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ ω) → ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹 ∘ 𝑁)‘𝑞) ≠ ((𝐹 ∘ 𝑁)‘𝑟))
69	68	ralrimiva 2570	. 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 4866	. . . . . . 7 ⊢ (𝑎 = 𝑥 → dom 𝑎 = dom 𝑥)
72	71	opeq1d 3814	. . . . . 6 ⊢ (𝑎 = 𝑥 → ⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩ = ⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩)
73	72	sneqd 3635	. . . . 5 ⊢ (𝑎 = 𝑥 → {⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩} = {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩})
74	70, 73	uneq12d 3318	. . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 ∪ {⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩}) = (𝑥 ∪ {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩}))
75	70, 74	ifeq12d 3580	. . 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 5558	. . . . 5 ⊢ (𝑏 = 𝑦 → ((𝐹 ∘ 𝑁)‘𝑏) = ((𝐹 ∘ 𝑁)‘𝑦))
77		imaeq2 5005	. . . . 5 ⊢ (𝑏 = 𝑦 → ((𝐹 ∘ 𝑁) “ 𝑏) = ((𝐹 ∘ 𝑁) “ 𝑦))
78	76, 77	eleq12d 2267	. . . 4 ⊢ (𝑏 = 𝑦 → (((𝐹 ∘ 𝑁)‘𝑏) ∈ ((𝐹 ∘ 𝑁) “ 𝑏) ↔ ((𝐹 ∘ 𝑁)‘𝑦) ∈ ((𝐹 ∘ 𝑁) “ 𝑦)))
79	76	opeq2d 3815	. . . . . 6 ⊢ (𝑏 = 𝑦 → ⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩ = ⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑦)⟩)
80	79	sneqd 3635	. . . . 5 ⊢ (𝑏 = 𝑦 → {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩} = {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑦)⟩})
81	80	uneq2d 3317	. . . 4 ⊢ (𝑏 = 𝑦 → (𝑥 ∪ {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩}) = (𝑥 ∪ {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑦)⟩}))
82	78, 81	ifbieq2d 3585	. . 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 6002	. 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 2203	. . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 = 0 ↔ 𝑥 = 0))
85		fvoveq1 5945	. . . 4 ⊢ (𝑎 = 𝑥 → (◡𝑁‘(𝑎 − 1)) = (◡𝑁‘(𝑥 − 1)))
86	84, 85	ifbieq2d 3585	. . 3 ⊢ (𝑎 = 𝑥 → if(𝑎 = 0, ∅, (◡𝑁‘(𝑎 − 1))) = if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))
87	86	cbvmptv 4129	. 2 ⊢ (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (◡𝑁‘(𝑎 − 1)))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))
88		eqid 2196	. 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 5558	. . 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 3955	. 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 12638	1 ⊢ (𝜑 → 𝐴 ≈ ℕ)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 835   = wceq 1364   ∈ wcel 2167   ≠ wne 2367  ∀wral 2475  ∃wrex 2476   ∪ cun 3155  ∅c0 3450  ifcif 3561  {csn 3622  ⟨cop 3625  ∪ ciun 3916   class class class wbr 4033   ↦ cmpt 4094  suc csuc 4400  ωcom 4626  ^◡ccnv 4662  dom cdm 4663   “ cima 4666   ∘ ccom 4667  ⟶wf 5254  –onto→wfo 5256  –1-1-onto→wf1o 5257  ‘cfv 5258  (class class class)co 5922   ∈ cmpo 5924  freccfrec 6448   ↑_pm cpm 6708   ≈ cen 6797  0cc0 7879  1c1 7880   + caddc 7882   < clt 8061   ≤ cle 8062   − cmin 8197  ℕcn 8990  ℕ₀cn0 9249  ℤcz 9326  ...cfz 10083  seqcseq 10539

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-er 6592  df-pm 6710  df-en 6800  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-seqfrec 10540

This theorem is referenced by:  ennnfonelemr  12640