Theorem ennnfonelemnn0 12579

Description: Lemma for ennnfone 12582. A version of ennnfonelemen 12578 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 10499	. . . . 5 ⊢ 𝑁:ω–1-1-onto→ℕ0
5		f1ofo 5507	. . . . 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 5487	. . 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 5926	. . . . . . 7 ⊢ (𝑛 = (𝑁‘𝑝) → (0...𝑛) = (0...(𝑁‘𝑝)))
11	10	raleqdv 2696	. . . . . 6 ⊢ (𝑛 = (𝑁‘𝑝) → (∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗)))
12	11	rexbidv 2495	. . . . 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 5500	. . . . . . . 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 5694	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ ω) → (𝑁‘𝑝) ∈ ℕ0)
20	12, 14, 19	rspcdva 2869	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ ω) → ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))
21		f1ocnv 5513	. . . . . . . 8 ⊢ (𝑁:ω–1-1-onto→ℕ0 → ◡𝑁:ℕ0–1-1-onto→ω)
22		f1of 5500	. . . . . . . 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 5694	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁‘𝑝))(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (◡𝑁‘𝑘) ∈ ω)
27		fveq2 5554	. . . . . . . . 9 ⊢ (𝑗 = (𝑁‘𝑟) → (𝐹‘𝑗) = (𝐹‘(𝑁‘𝑟)))
28	27	neeq2d 2383	. . . . . . . 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 4627	. . . . . . . . . . . 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 4638	. . . . . . . . . . 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 5692	. . . . . . . . . 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 9329	. . . . . . . . . . . . 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 10474	. . . . . . . . . . . 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 10472	. . . . . . . . . . 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 4055	. . . . . . . . . 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 9380	. . . . . . . . . . 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 10179	. . . . . . . . . 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 2869	. . . . . . 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 5628	. . . . . . . . 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 5821	. . . . . . . . . 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 5558	. . . . . . . 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 2226	. . . . . . 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 5628	. . . . . . . 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 2397	. . . . . 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 2567	. . . . 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 5554	. . . . . . . 8 ⊢ (𝑞 = (◡𝑁‘𝑘) → ((𝐹 ∘ 𝑁)‘𝑞) = ((𝐹 ∘ 𝑁)‘(◡𝑁‘𝑘)))
64	63	neeq1d 2382	. . . . . . 7 ⊢ (𝑞 = (◡𝑁‘𝑘) → (((𝐹 ∘ 𝑁)‘𝑞) ≠ ((𝐹 ∘ 𝑁)‘𝑟) ↔ ((𝐹 ∘ 𝑁)‘(◡𝑁‘𝑘)) ≠ ((𝐹 ∘ 𝑁)‘𝑟)))
65	64	ralbidv 2494	. . . . . 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 2864	. . . . 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 2616	. . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ ω) → ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹 ∘ 𝑁)‘𝑞) ≠ ((𝐹 ∘ 𝑁)‘𝑟))
69	68	ralrimiva 2567	. 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 4862	. . . . . . 7 ⊢ (𝑎 = 𝑥 → dom 𝑎 = dom 𝑥)
72	71	opeq1d 3810	. . . . . 6 ⊢ (𝑎 = 𝑥 → ⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩ = ⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩)
73	72	sneqd 3631	. . . . 5 ⊢ (𝑎 = 𝑥 → {⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩} = {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩})
74	70, 73	uneq12d 3314	. . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 ∪ {⟨dom 𝑎, ((𝐹 ∘ 𝑁)‘𝑏)⟩}) = (𝑥 ∪ {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩}))
75	70, 74	ifeq12d 3576	. . 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 5554	. . . . 5 ⊢ (𝑏 = 𝑦 → ((𝐹 ∘ 𝑁)‘𝑏) = ((𝐹 ∘ 𝑁)‘𝑦))
77		imaeq2 5001	. . . . 5 ⊢ (𝑏 = 𝑦 → ((𝐹 ∘ 𝑁) “ 𝑏) = ((𝐹 ∘ 𝑁) “ 𝑦))
78	76, 77	eleq12d 2264	. . . 4 ⊢ (𝑏 = 𝑦 → (((𝐹 ∘ 𝑁)‘𝑏) ∈ ((𝐹 ∘ 𝑁) “ 𝑏) ↔ ((𝐹 ∘ 𝑁)‘𝑦) ∈ ((𝐹 ∘ 𝑁) “ 𝑦)))
79	76	opeq2d 3811	. . . . . 6 ⊢ (𝑏 = 𝑦 → ⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩ = ⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑦)⟩)
80	79	sneqd 3631	. . . . 5 ⊢ (𝑏 = 𝑦 → {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩} = {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑦)⟩})
81	80	uneq2d 3313	. . . 4 ⊢ (𝑏 = 𝑦 → (𝑥 ∪ {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑏)⟩}) = (𝑥 ∪ {⟨dom 𝑥, ((𝐹 ∘ 𝑁)‘𝑦)⟩}))
82	78, 81	ifbieq2d 3581	. . 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 5998	. 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 2200	. . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 = 0 ↔ 𝑥 = 0))
85		fvoveq1 5941	. . . 4 ⊢ (𝑎 = 𝑥 → (◡𝑁‘(𝑎 − 1)) = (◡𝑁‘(𝑥 − 1)))
86	84, 85	ifbieq2d 3581	. . 3 ⊢ (𝑎 = 𝑥 → if(𝑎 = 0, ∅, (◡𝑁‘(𝑎 − 1))) = if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))
87	86	cbvmptv 4125	. 2 ⊢ (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (◡𝑁‘(𝑎 − 1)))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))
88		eqid 2193	. 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 5554	. . 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 3951	. 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 12578	1 ⊢ (𝜑 → 𝐴 ≈ ℕ)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 835   = wceq 1364   ∈ wcel 2164   ≠ wne 2364  ∀wral 2472  ∃wrex 2473   ∪ cun 3151  ∅c0 3446  ifcif 3557  {csn 3618  ⟨cop 3621  ∪ ciun 3912   class class class wbr 4029   ↦ cmpt 4090  suc csuc 4396  ωcom 4622  ^◡ccnv 4658  dom cdm 4659   “ cima 4662   ∘ ccom 4663  ⟶wf 5250  –onto→wfo 5252  –1-1-onto→wf1o 5253  ‘cfv 5254  (class class class)co 5918   ∈ cmpo 5920  freccfrec 6443   ↑_pm cpm 6703   ≈ cen 6792  0cc0 7872  1c1 7873   + caddc 7875   < clt 8054   ≤ cle 8055   − cmin 8190  ℕcn 8982  ℕ₀cn0 9240  ℤcz 9317  ...cfz 10074  seqcseq 10518

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-er 6587  df-pm 6705  df-en 6795  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-seqfrec 10519

This theorem is referenced by:  ennnfonelemr  12580