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Theorem ennnfonelemnn0 11958
 Description: Lemma for ennnfone 11961. A version of ennnfonelemen 11957 expressed in terms of ℕ0 instead of ω. (Contributed by Jim Kingdon, 27-Oct-2022.)
Hypotheses
Ref Expression
ennnfonelemr.dceq (𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
ennnfonelemr.f (𝜑𝐹:ℕ0onto𝐴)
ennnfonelemr.n (𝜑 → ∀𝑛 ∈ ℕ0𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝐹𝑘) ≠ (𝐹𝑗))
ennnfonelemnn0.n 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
Assertion
Ref Expression
ennnfonelemnn0 (𝜑𝐴 ≈ 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.
StepHypRef Expression
1 ennnfonelemr.dceq . 2 (𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
2 ennnfonelemr.f . . 3 (𝜑𝐹:ℕ0onto𝐴)
3 ennnfonelemnn0.n . . . . . 6 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
43frechashgf1o 10225 . . . . 5 𝑁:ω–1-1-onto→ℕ0
5 f1ofo 5377 . . . . 5 (𝑁:ω–1-1-onto→ℕ0𝑁:ω–onto→ℕ0)
64, 5ax-mp 5 . . . 4 𝑁:ω–onto→ℕ0
76a1i 9 . . 3 (𝜑𝑁:ω–onto→ℕ0)
8 foco 5358 . . 3 ((𝐹:ℕ0onto𝐴𝑁:ω–onto→ℕ0) → (𝐹𝑁):ω–onto𝐴)
92, 7, 8syl2anc 408 . 2 (𝜑 → (𝐹𝑁):ω–onto𝐴)
10 oveq2 5785 . . . . . . 7 (𝑛 = (𝑁𝑝) → (0...𝑛) = (0...(𝑁𝑝)))
1110raleqdv 2632 . . . . . 6 (𝑛 = (𝑁𝑝) → (∀𝑗 ∈ (0...𝑛)(𝐹𝑘) ≠ (𝐹𝑗) ↔ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗)))
1211rexbidv 2438 . . . . 5 (𝑛 = (𝑁𝑝) → (∃𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝐹𝑘) ≠ (𝐹𝑗) ↔ ∃𝑘 ∈ ℕ0𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗)))
13 ennnfonelemr.n . . . . . 6 (𝜑 → ∀𝑛 ∈ ℕ0𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝐹𝑘) ≠ (𝐹𝑗))
1413adantr 274 . . . . 5 ((𝜑𝑝 ∈ ω) → ∀𝑛 ∈ ℕ0𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝐹𝑘) ≠ (𝐹𝑗))
15 f1of 5370 . . . . . . . 8 (𝑁:ω–1-1-onto→ℕ0𝑁:ω⟶ℕ0)
164, 15ax-mp 5 . . . . . . 7 𝑁:ω⟶ℕ0
1716a1i 9 . . . . . 6 ((𝜑𝑝 ∈ ω) → 𝑁:ω⟶ℕ0)
18 simpr 109 . . . . . 6 ((𝜑𝑝 ∈ ω) → 𝑝 ∈ ω)
1917, 18ffvelrnd 5559 . . . . 5 ((𝜑𝑝 ∈ ω) → (𝑁𝑝) ∈ ℕ0)
2012, 14, 19rspcdva 2794 . . . 4 ((𝜑𝑝 ∈ ω) → ∃𝑘 ∈ ℕ0𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))
21 f1ocnv 5383 . . . . . . . 8 (𝑁:ω–1-1-onto→ℕ0𝑁:ℕ01-1-onto→ω)
22 f1of 5370 . . . . . . . 8 (𝑁:ℕ01-1-onto→ω → 𝑁:ℕ0⟶ω)
234, 21, 22mp2b 8 . . . . . . 7 𝑁:ℕ0⟶ω
2423a1i 9 . . . . . 6 (((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) → 𝑁:ℕ0⟶ω)
25 simprl 520 . . . . . 6 (((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) → 𝑘 ∈ ℕ0)
2624, 25ffvelrnd 5559 . . . . 5 (((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) → (𝑁𝑘) ∈ ω)
27 fveq2 5424 . . . . . . . . 9 (𝑗 = (𝑁𝑟) → (𝐹𝑗) = (𝐹‘(𝑁𝑟)))
2827neeq2d 2327 . . . . . . . 8 (𝑗 = (𝑁𝑟) → ((𝐹𝑘) ≠ (𝐹𝑗) ↔ (𝐹𝑘) ≠ (𝐹‘(𝑁𝑟))))
29 simplrr 525 . . . . . . . 8 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))
30 simpr 109 . . . . . . . . . . 11 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑟 ∈ suc 𝑝)
3118ad2antrr 479 . . . . . . . . . . . 12 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑝 ∈ ω)
32 peano2 4512 . . . . . . . . . . . 12 (𝑝 ∈ ω → suc 𝑝 ∈ ω)
3331, 32syl 14 . . . . . . . . . . 11 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → suc 𝑝 ∈ ω)
34 elnn 4522 . . . . . . . . . . 11 ((𝑟 ∈ suc 𝑝 ∧ suc 𝑝 ∈ ω) → 𝑟 ∈ ω)
3530, 33, 34syl2anc 408 . . . . . . . . . 10 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑟 ∈ ω)
3616ffvelrni 5557 . . . . . . . . . 10 (𝑟 ∈ ω → (𝑁𝑟) ∈ ℕ0)
3735, 36syl 14 . . . . . . . . 9 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑟) ∈ ℕ0)
38 0zd 9085 . . . . . . . . . . . . 13 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 0 ∈ ℤ)
3938, 3, 35, 33frec2uzltd 10200 . . . . . . . . . . . 12 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑟 ∈ suc 𝑝 → (𝑁𝑟) < (𝑁‘suc 𝑝)))
4030, 39mpd 13 . . . . . . . . . . 11 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑟) < (𝑁‘suc 𝑝))
4138, 3, 31frec2uzsucd 10198 . . . . . . . . . . 11 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘suc 𝑝) = ((𝑁𝑝) + 1))
4240, 41breqtrd 3957 . . . . . . . . . 10 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑟) < ((𝑁𝑝) + 1))
4319ad2antrr 479 . . . . . . . . . . 11 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑝) ∈ ℕ0)
44 nn0leltp1 9136 . . . . . . . . . . 11 (((𝑁𝑟) ∈ ℕ0 ∧ (𝑁𝑝) ∈ ℕ0) → ((𝑁𝑟) ≤ (𝑁𝑝) ↔ (𝑁𝑟) < ((𝑁𝑝) + 1)))
4537, 43, 44syl2anc 408 . . . . . . . . . 10 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝑁𝑟) ≤ (𝑁𝑝) ↔ (𝑁𝑟) < ((𝑁𝑝) + 1)))
4642, 45mpbird 166 . . . . . . . . 9 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑟) ≤ (𝑁𝑝))
47 fznn0 9917 . . . . . . . . . 10 ((𝑁𝑝) ∈ ℕ0 → ((𝑁𝑟) ∈ (0...(𝑁𝑝)) ↔ ((𝑁𝑟) ∈ ℕ0 ∧ (𝑁𝑟) ≤ (𝑁𝑝))))
4843, 47syl 14 . . . . . . . . 9 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝑁𝑟) ∈ (0...(𝑁𝑝)) ↔ ((𝑁𝑟) ∈ ℕ0 ∧ (𝑁𝑟) ≤ (𝑁𝑝))))
4937, 46, 48mpbir2and 928 . . . . . . . 8 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑟) ∈ (0...(𝑁𝑝)))
5028, 29, 49rspcdva 2794 . . . . . . 7 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝐹𝑘) ≠ (𝐹‘(𝑁𝑟)))
5126adantr 274 . . . . . . . . 9 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑘) ∈ ω)
52 fvco3 5495 . . . . . . . . 9 ((𝑁:ω⟶ℕ0 ∧ (𝑁𝑘) ∈ ω) → ((𝐹𝑁)‘(𝑁𝑘)) = (𝐹‘(𝑁‘(𝑁𝑘))))
5316, 51, 52sylancr 410 . . . . . . . 8 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹𝑁)‘(𝑁𝑘)) = (𝐹‘(𝑁‘(𝑁𝑘))))
5425adantr 274 . . . . . . . . . 10 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑘 ∈ ℕ0)
55 f1ocnvfv2 5682 . . . . . . . . . 10 ((𝑁:ω–1-1-onto→ℕ0𝑘 ∈ ℕ0) → (𝑁‘(𝑁𝑘)) = 𝑘)
564, 54, 55sylancr 410 . . . . . . . . 9 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘(𝑁𝑘)) = 𝑘)
5756fveq2d 5428 . . . . . . . 8 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝐹‘(𝑁‘(𝑁𝑘))) = (𝐹𝑘))
5853, 57eqtrd 2172 . . . . . . 7 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹𝑁)‘(𝑁𝑘)) = (𝐹𝑘))
59 fvco3 5495 . . . . . . . 8 ((𝑁:ω⟶ℕ0𝑟 ∈ ω) → ((𝐹𝑁)‘𝑟) = (𝐹‘(𝑁𝑟)))
6016, 35, 59sylancr 410 . . . . . . 7 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹𝑁)‘𝑟) = (𝐹‘(𝑁𝑟)))
6150, 58, 603netr4d 2341 . . . . . 6 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟))
6261ralrimiva 2505 . . . . 5 (((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) → ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟))
63 fveq2 5424 . . . . . . . 8 (𝑞 = (𝑁𝑘) → ((𝐹𝑁)‘𝑞) = ((𝐹𝑁)‘(𝑁𝑘)))
6463neeq1d 2326 . . . . . . 7 (𝑞 = (𝑁𝑘) → (((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟) ↔ ((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟)))
6564ralbidv 2437 . . . . . 6 (𝑞 = (𝑁𝑘) → (∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟) ↔ ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟)))
6665rspcev 2789 . . . . 5 (((𝑁𝑘) ∈ ω ∧ ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟)) → ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟))
6726, 62, 66syl2anc 408 . . . 4 (((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) → ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟))
6820, 67rexlimddv 2554 . . 3 ((𝜑𝑝 ∈ ω) → ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟))
6968ralrimiva 2505 . 2 (𝜑 → ∀𝑝 ∈ ω ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟))
70 id 19 . . . 4 (𝑎 = 𝑥𝑎 = 𝑥)
71 dmeq 4742 . . . . . . 7 (𝑎 = 𝑥 → dom 𝑎 = dom 𝑥)
7271opeq1d 3714 . . . . . 6 (𝑎 = 𝑥 → ⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩ = ⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩)
7372sneqd 3540 . . . . 5 (𝑎 = 𝑥 → {⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩} = {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩})
7470, 73uneq12d 3231 . . . 4 (𝑎 = 𝑥 → (𝑎 ∪ {⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩}) = (𝑥 ∪ {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩}))
7570, 74ifeq12d 3491 . . 3 (𝑎 = 𝑥 → if(((𝐹𝑁)‘𝑏) ∈ ((𝐹𝑁) “ 𝑏), 𝑎, (𝑎 ∪ {⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩})) = if(((𝐹𝑁)‘𝑏) ∈ ((𝐹𝑁) “ 𝑏), 𝑥, (𝑥 ∪ {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩})))
76 fveq2 5424 . . . . 5 (𝑏 = 𝑦 → ((𝐹𝑁)‘𝑏) = ((𝐹𝑁)‘𝑦))
77 imaeq2 4880 . . . . 5 (𝑏 = 𝑦 → ((𝐹𝑁) “ 𝑏) = ((𝐹𝑁) “ 𝑦))
7876, 77eleq12d 2210 . . . 4 (𝑏 = 𝑦 → (((𝐹𝑁)‘𝑏) ∈ ((𝐹𝑁) “ 𝑏) ↔ ((𝐹𝑁)‘𝑦) ∈ ((𝐹𝑁) “ 𝑦)))
7976opeq2d 3715 . . . . . 6 (𝑏 = 𝑦 → ⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩ = ⟨dom 𝑥, ((𝐹𝑁)‘𝑦)⟩)
8079sneqd 3540 . . . . 5 (𝑏 = 𝑦 → {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩} = {⟨dom 𝑥, ((𝐹𝑁)‘𝑦)⟩})
8180uneq2d 3230 . . . 4 (𝑏 = 𝑦 → (𝑥 ∪ {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩}) = (𝑥 ∪ {⟨dom 𝑥, ((𝐹𝑁)‘𝑦)⟩}))
8278, 81ifbieq2d 3496 . . 3 (𝑏 = 𝑦 → if(((𝐹𝑁)‘𝑏) ∈ ((𝐹𝑁) “ 𝑏), 𝑥, (𝑥 ∪ {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩})) = if(((𝐹𝑁)‘𝑦) ∈ ((𝐹𝑁) “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, ((𝐹𝑁)‘𝑦)⟩})))
8375, 82cbvmpov 5854 . 2 (𝑎 ∈ (𝐴pm ω), 𝑏 ∈ ω ↦ if(((𝐹𝑁)‘𝑏) ∈ ((𝐹𝑁) “ 𝑏), 𝑎, (𝑎 ∪ {⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩}))) = (𝑥 ∈ (𝐴pm ω), 𝑦 ∈ ω ↦ if(((𝐹𝑁)‘𝑦) ∈ ((𝐹𝑁) “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, ((𝐹𝑁)‘𝑦)⟩})))
84 eqeq1 2146 . . . 4 (𝑎 = 𝑥 → (𝑎 = 0 ↔ 𝑥 = 0))
85 fvoveq1 5800 . . . 4 (𝑎 = 𝑥 → (𝑁‘(𝑎 − 1)) = (𝑁‘(𝑥 − 1)))
8684, 85ifbieq2d 3496 . . 3 (𝑎 = 𝑥 → if(𝑎 = 0, ∅, (𝑁‘(𝑎 − 1))) = if(𝑥 = 0, ∅, (𝑁‘(𝑥 − 1))))
8786cbvmptv 4027 . 2 (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (𝑁‘(𝑎 − 1)))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (𝑁‘(𝑥 − 1))))
88 eqid 2139 . 2 seq0((𝑎 ∈ (𝐴pm ω), 𝑏 ∈ ω ↦ if(((𝐹𝑁)‘𝑏) ∈ ((𝐹𝑁) “ 𝑏), 𝑎, (𝑎 ∪ {⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩}))), (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (𝑁‘(𝑎 − 1))))) = seq0((𝑎 ∈ (𝐴pm ω), 𝑏 ∈ ω ↦ if(((𝐹𝑁)‘𝑏) ∈ ((𝐹𝑁) “ 𝑏), 𝑎, (𝑎 ∪ {⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩}))), (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (𝑁‘(𝑎 − 1)))))
89 fveq2 5424 . . 3 (𝑖 = 𝑐 → (seq0((𝑎 ∈ (𝐴pm ω), 𝑏 ∈ ω ↦ if(((𝐹𝑁)‘𝑏) ∈ ((𝐹𝑁) “ 𝑏), 𝑎, (𝑎 ∪ {⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩}))), (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (𝑁‘(𝑎 − 1)))))‘𝑖) = (seq0((𝑎 ∈ (𝐴pm ω), 𝑏 ∈ ω ↦ if(((𝐹𝑁)‘𝑏) ∈ ((𝐹𝑁) “ 𝑏), 𝑎, (𝑎 ∪ {⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩}))), (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (𝑁‘(𝑎 − 1)))))‘𝑐))
9089cbviunv 3855 . 2 𝑖 ∈ ℕ0 (seq0((𝑎 ∈ (𝐴pm ω), 𝑏 ∈ ω ↦ if(((𝐹𝑁)‘𝑏) ∈ ((𝐹𝑁) “ 𝑏), 𝑎, (𝑎 ∪ {⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩}))), (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (𝑁‘(𝑎 − 1)))))‘𝑖) = 𝑐 ∈ ℕ0 (seq0((𝑎 ∈ (𝐴pm ω), 𝑏 ∈ ω ↦ if(((𝐹𝑁)‘𝑏) ∈ ((𝐹𝑁) “ 𝑏), 𝑎, (𝑎 ∪ {⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩}))), (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (𝑁‘(𝑎 − 1)))))‘𝑐)
911, 9, 69, 83, 3, 87, 88, 90ennnfonelemen 11957 1 (𝜑𝐴 ≈ ℕ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 819   = wceq 1331   ∈ wcel 1480   ≠ wne 2308  ∀wral 2416  ∃wrex 2417   ∪ cun 3069  ∅c0 3363  ifcif 3474  {csn 3527  ⟨cop 3530  ∪ ciun 3816   class class class wbr 3932   ↦ cmpt 3992  suc csuc 4290  ωcom 4507  ◡ccnv 4541  dom cdm 4542   “ cima 4545   ∘ ccom 4546  ⟶wf 5122  –onto→wfo 5124  –1-1-onto→wf1o 5125  ‘cfv 5126  (class class class)co 5777   ∈ cmpo 5779  freccfrec 6290   ↑pm cpm 6546   ≈ cen 6635  0cc0 7639  1c1 7640   + caddc 7642   < clt 7819   ≤ cle 7820   − cmin 7952  ℕcn 8739  ℕ0cn0 8996  ℤcz 9073  ...cfz 9814  seqcseq 10242 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4046  ax-sep 4049  ax-nul 4057  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455  ax-iinf 4505  ax-cnex 7730  ax-resscn 7731  ax-1cn 7732  ax-1re 7733  ax-icn 7734  ax-addcl 7735  ax-addrcl 7736  ax-mulcl 7737  ax-addcom 7739  ax-addass 7741  ax-distr 7743  ax-i2m1 7744  ax-0lt1 7745  ax-0id 7747  ax-rnegex 7748  ax-cnre 7750  ax-pre-ltirr 7751  ax-pre-ltwlin 7752  ax-pre-lttrn 7753  ax-pre-ltadd 7755 This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-int 3775  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-tr 4030  df-id 4218  df-iord 4291  df-on 4293  df-ilim 4294  df-suc 4296  df-iom 4508  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133  df-fv 5134  df-riota 5733  df-ov 5780  df-oprab 5781  df-mpo 5782  df-1st 6041  df-2nd 6042  df-recs 6205  df-frec 6291  df-er 6432  df-pm 6548  df-en 6638  df-pnf 7821  df-mnf 7822  df-xr 7823  df-ltxr 7824  df-le 7825  df-sub 7954  df-neg 7955  df-inn 8740  df-n0 8997  df-z 9074  df-uz 9346  df-fz 9815  df-seqfrec 10243 This theorem is referenced by:  ennnfonelemr  11959
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