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Theorem ennnfonelemnn0 11958
Description: Lemma for ennnfone 11961. A version of ennnfonelemen 11957 expressed in terms of 0 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  ontowfo 5124  1-1-ontowf1o 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  cn 8739  0cn0 8996  cz 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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