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Theorem ennnfonelemnn0 11780
Description: Lemma for ennnfone 11783. A version of ennnfonelemen 11779 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 10094 . . . . 5 𝑁:ω–1-1-onto→ℕ0
5 f1ofo 5330 . . . . 5 (𝑁:ω–1-1-onto→ℕ0𝑁:ω–onto→ℕ0)
64, 5ax-mp 7 . . . 4 𝑁:ω–onto→ℕ0
76a1i 9 . . 3 (𝜑𝑁:ω–onto→ℕ0)
8 foco 5313 . . 3 ((𝐹:ℕ0onto𝐴𝑁:ω–onto→ℕ0) → (𝐹𝑁):ω–onto𝐴)
92, 7, 8syl2anc 406 . 2 (𝜑 → (𝐹𝑁):ω–onto𝐴)
10 oveq2 5736 . . . . . . 7 (𝑛 = (𝑁𝑝) → (0...𝑛) = (0...(𝑁𝑝)))
1110raleqdv 2606 . . . . . 6 (𝑛 = (𝑁𝑝) → (∀𝑗 ∈ (0...𝑛)(𝐹𝑘) ≠ (𝐹𝑗) ↔ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗)))
1211rexbidv 2412 . . . . 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 272 . . . . 5 ((𝜑𝑝 ∈ ω) → ∀𝑛 ∈ ℕ0𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝐹𝑘) ≠ (𝐹𝑗))
15 f1of 5323 . . . . . . . 8 (𝑁:ω–1-1-onto→ℕ0𝑁:ω⟶ℕ0)
164, 15ax-mp 7 . . . . . . 7 𝑁:ω⟶ℕ0
1716a1i 9 . . . . . 6 ((𝜑𝑝 ∈ ω) → 𝑁:ω⟶ℕ0)
18 simpr 109 . . . . . 6 ((𝜑𝑝 ∈ ω) → 𝑝 ∈ ω)
1917, 18ffvelrnd 5510 . . . . 5 ((𝜑𝑝 ∈ ω) → (𝑁𝑝) ∈ ℕ0)
2012, 14, 19rspcdva 2765 . . . 4 ((𝜑𝑝 ∈ ω) → ∃𝑘 ∈ ℕ0𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))
21 f1ocnv 5336 . . . . . . . 8 (𝑁:ω–1-1-onto→ℕ0𝑁:ℕ01-1-onto→ω)
22 f1of 5323 . . . . . . . 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 503 . . . . . 6 (((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) → 𝑘 ∈ ℕ0)
2624, 25ffvelrnd 5510 . . . . 5 (((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) → (𝑁𝑘) ∈ ω)
27 fveq2 5375 . . . . . . . . 9 (𝑗 = (𝑁𝑟) → (𝐹𝑗) = (𝐹‘(𝑁𝑟)))
2827neeq2d 2301 . . . . . . . 8 (𝑗 = (𝑁𝑟) → ((𝐹𝑘) ≠ (𝐹𝑗) ↔ (𝐹𝑘) ≠ (𝐹‘(𝑁𝑟))))
29 simplrr 508 . . . . . . . 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 477 . . . . . . . . . . . 12 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑝 ∈ ω)
32 peano2 4469 . . . . . . . . . . . 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 4479 . . . . . . . . . . 11 ((𝑟 ∈ suc 𝑝 ∧ suc 𝑝 ∈ ω) → 𝑟 ∈ ω)
3530, 33, 34syl2anc 406 . . . . . . . . . 10 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑟 ∈ ω)
3616ffvelrni 5508 . . . . . . . . . 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 8970 . . . . . . . . . . . . 13 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 0 ∈ ℤ)
3938, 3, 35, 33frec2uzltd 10069 . . . . . . . . . . . 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 10067 . . . . . . . . . . 11 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘suc 𝑝) = ((𝑁𝑝) + 1))
4240, 41breqtrd 3919 . . . . . . . . . 10 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑟) < ((𝑁𝑝) + 1))
4319ad2antrr 477 . . . . . . . . . . 11 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑝) ∈ ℕ0)
44 nn0leltp1 9021 . . . . . . . . . . 11 (((𝑁𝑟) ∈ ℕ0 ∧ (𝑁𝑝) ∈ ℕ0) → ((𝑁𝑟) ≤ (𝑁𝑝) ↔ (𝑁𝑟) < ((𝑁𝑝) + 1)))
4537, 43, 44syl2anc 406 . . . . . . . . . 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 9786 . . . . . . . . . 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 911 . . . . . . . 8 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑟) ∈ (0...(𝑁𝑝)))
5028, 29, 49rspcdva 2765 . . . . . . 7 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝐹𝑘) ≠ (𝐹‘(𝑁𝑟)))
5126adantr 272 . . . . . . . . 9 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑘) ∈ ω)
52 fvco3 5446 . . . . . . . . 9 ((𝑁:ω⟶ℕ0 ∧ (𝑁𝑘) ∈ ω) → ((𝐹𝑁)‘(𝑁𝑘)) = (𝐹‘(𝑁‘(𝑁𝑘))))
5316, 51, 52sylancr 408 . . . . . . . 8 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹𝑁)‘(𝑁𝑘)) = (𝐹‘(𝑁‘(𝑁𝑘))))
5425adantr 272 . . . . . . . . . 10 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑘 ∈ ℕ0)
55 f1ocnvfv2 5633 . . . . . . . . . 10 ((𝑁:ω–1-1-onto→ℕ0𝑘 ∈ ℕ0) → (𝑁‘(𝑁𝑘)) = 𝑘)
564, 54, 55sylancr 408 . . . . . . . . 9 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘(𝑁𝑘)) = 𝑘)
5756fveq2d 5379 . . . . . . . 8 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝐹‘(𝑁‘(𝑁𝑘))) = (𝐹𝑘))
5853, 57eqtrd 2147 . . . . . . 7 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹𝑁)‘(𝑁𝑘)) = (𝐹𝑘))
59 fvco3 5446 . . . . . . . 8 ((𝑁:ω⟶ℕ0𝑟 ∈ ω) → ((𝐹𝑁)‘𝑟) = (𝐹‘(𝑁𝑟)))
6016, 35, 59sylancr 408 . . . . . . 7 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹𝑁)‘𝑟) = (𝐹‘(𝑁𝑟)))
6150, 58, 603netr4d 2315 . . . . . 6 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟))
6261ralrimiva 2479 . . . . 5 (((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) → ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟))
63 fveq2 5375 . . . . . . . 8 (𝑞 = (𝑁𝑘) → ((𝐹𝑁)‘𝑞) = ((𝐹𝑁)‘(𝑁𝑘)))
6463neeq1d 2300 . . . . . . 7 (𝑞 = (𝑁𝑘) → (((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟) ↔ ((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟)))
6564ralbidv 2411 . . . . . 6 (𝑞 = (𝑁𝑘) → (∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟) ↔ ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟)))
6665rspcev 2760 . . . . 5 (((𝑁𝑘) ∈ ω ∧ ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟)) → ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟))
6726, 62, 66syl2anc 406 . . . 4 (((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) → ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟))
6820, 67rexlimddv 2528 . . 3 ((𝜑𝑝 ∈ ω) → ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟))
6968ralrimiva 2479 . 2 (𝜑 → ∀𝑝 ∈ ω ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟))
70 id 19 . . . 4 (𝑎 = 𝑥𝑎 = 𝑥)
71 dmeq 4699 . . . . . . 7 (𝑎 = 𝑥 → dom 𝑎 = dom 𝑥)
7271opeq1d 3677 . . . . . 6 (𝑎 = 𝑥 → ⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩ = ⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩)
7372sneqd 3506 . . . . 5 (𝑎 = 𝑥 → {⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩} = {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩})
7470, 73uneq12d 3197 . . . 4 (𝑎 = 𝑥 → (𝑎 ∪ {⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩}) = (𝑥 ∪ {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩}))
7570, 74ifeq12d 3457 . . 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 5375 . . . . 5 (𝑏 = 𝑦 → ((𝐹𝑁)‘𝑏) = ((𝐹𝑁)‘𝑦))
77 imaeq2 4835 . . . . 5 (𝑏 = 𝑦 → ((𝐹𝑁) “ 𝑏) = ((𝐹𝑁) “ 𝑦))
7876, 77eleq12d 2185 . . . 4 (𝑏 = 𝑦 → (((𝐹𝑁)‘𝑏) ∈ ((𝐹𝑁) “ 𝑏) ↔ ((𝐹𝑁)‘𝑦) ∈ ((𝐹𝑁) “ 𝑦)))
7976opeq2d 3678 . . . . . 6 (𝑏 = 𝑦 → ⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩ = ⟨dom 𝑥, ((𝐹𝑁)‘𝑦)⟩)
8079sneqd 3506 . . . . 5 (𝑏 = 𝑦 → {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩} = {⟨dom 𝑥, ((𝐹𝑁)‘𝑦)⟩})
8180uneq2d 3196 . . . 4 (𝑏 = 𝑦 → (𝑥 ∪ {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩}) = (𝑥 ∪ {⟨dom 𝑥, ((𝐹𝑁)‘𝑦)⟩}))
8278, 81ifbieq2d 3462 . . 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 5805 . 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 2121 . . . 4 (𝑎 = 𝑥 → (𝑎 = 0 ↔ 𝑥 = 0))
85 fvoveq1 5751 . . . 4 (𝑎 = 𝑥 → (𝑁‘(𝑎 − 1)) = (𝑁‘(𝑥 − 1)))
8684, 85ifbieq2d 3462 . . 3 (𝑎 = 𝑥 → if(𝑎 = 0, ∅, (𝑁‘(𝑎 − 1))) = if(𝑥 = 0, ∅, (𝑁‘(𝑥 − 1))))
8786cbvmptv 3984 . 2 (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (𝑁‘(𝑎 − 1)))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (𝑁‘(𝑥 − 1))))
88 eqid 2115 . 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 5375 . . 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 3818 . 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 11779 1 (𝜑𝐴 ≈ ℕ)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  DECID wdc 802   = wceq 1314  wcel 1463  wne 2282  wral 2390  wrex 2391  cun 3035  c0 3329  ifcif 3440  {csn 3493  cop 3496   ciun 3779   class class class wbr 3895  cmpt 3949  suc csuc 4247  ωcom 4464  ccnv 4498  dom cdm 4499  cima 4502  ccom 4503  wf 5077  ontowfo 5079  1-1-ontowf1o 5080  cfv 5081  (class class class)co 5728  cmpo 5730  freccfrec 6241  pm cpm 6497  cen 6586  0cc0 7547  1c1 7548   + caddc 7550   < clt 7724  cle 7725  cmin 7856  cn 8630  0cn0 8881  cz 8958  ...cfz 9683  seqcseq 10111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-addcom 7645  ax-addass 7647  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-0id 7653  ax-rnegex 7654  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-ltadd 7661
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-er 6383  df-pm 6499  df-en 6589  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-inn 8631  df-n0 8882  df-z 8959  df-uz 9229  df-fz 9684  df-seqfrec 10112
This theorem is referenced by:  ennnfonelemr  11781
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