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Theorem ennnfonelemnn0 12390
Description: Lemma for ennnfone 12393. A version of ennnfonelemen 12389 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 10398 . . . . 5 𝑁:ω–1-1-onto→ℕ0
5 f1ofo 5460 . . . . 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 5440 . . 3 ((𝐹:ℕ0onto𝐴𝑁:ω–onto→ℕ0) → (𝐹𝑁):ω–onto𝐴)
92, 7, 8syl2anc 411 . 2 (𝜑 → (𝐹𝑁):ω–onto𝐴)
10 oveq2 5873 . . . . . . 7 (𝑛 = (𝑁𝑝) → (0...𝑛) = (0...(𝑁𝑝)))
1110raleqdv 2676 . . . . . 6 (𝑛 = (𝑁𝑝) → (∀𝑗 ∈ (0...𝑛)(𝐹𝑘) ≠ (𝐹𝑗) ↔ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗)))
1211rexbidv 2476 . . . . 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 276 . . . . 5 ((𝜑𝑝 ∈ ω) → ∀𝑛 ∈ ℕ0𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝐹𝑘) ≠ (𝐹𝑗))
15 f1of 5453 . . . . . . . 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 110 . . . . . 6 ((𝜑𝑝 ∈ ω) → 𝑝 ∈ ω)
1917, 18ffvelcdmd 5644 . . . . 5 ((𝜑𝑝 ∈ ω) → (𝑁𝑝) ∈ ℕ0)
2012, 14, 19rspcdva 2844 . . . 4 ((𝜑𝑝 ∈ ω) → ∃𝑘 ∈ ℕ0𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))
21 f1ocnv 5466 . . . . . . . 8 (𝑁:ω–1-1-onto→ℕ0𝑁:ℕ01-1-onto→ω)
22 f1of 5453 . . . . . . . 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 529 . . . . . 6 (((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) → 𝑘 ∈ ℕ0)
2624, 25ffvelcdmd 5644 . . . . 5 (((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) → (𝑁𝑘) ∈ ω)
27 fveq2 5507 . . . . . . . . 9 (𝑗 = (𝑁𝑟) → (𝐹𝑗) = (𝐹‘(𝑁𝑟)))
2827neeq2d 2364 . . . . . . . 8 (𝑗 = (𝑁𝑟) → ((𝐹𝑘) ≠ (𝐹𝑗) ↔ (𝐹𝑘) ≠ (𝐹‘(𝑁𝑟))))
29 simplrr 536 . . . . . . . 8 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))
30 simpr 110 . . . . . . . . . . 11 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑟 ∈ suc 𝑝)
3118ad2antrr 488 . . . . . . . . . . . 12 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑝 ∈ ω)
32 peano2 4588 . . . . . . . . . . . 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 4599 . . . . . . . . . . 11 ((𝑟 ∈ suc 𝑝 ∧ suc 𝑝 ∈ ω) → 𝑟 ∈ ω)
3530, 33, 34syl2anc 411 . . . . . . . . . 10 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑟 ∈ ω)
3616ffvelcdmi 5642 . . . . . . . . . 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 9238 . . . . . . . . . . . . 13 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 0 ∈ ℤ)
3938, 3, 35, 33frec2uzltd 10373 . . . . . . . . . . . 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 10371 . . . . . . . . . . 11 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘suc 𝑝) = ((𝑁𝑝) + 1))
4240, 41breqtrd 4024 . . . . . . . . . 10 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑟) < ((𝑁𝑝) + 1))
4319ad2antrr 488 . . . . . . . . . . 11 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑝) ∈ ℕ0)
44 nn0leltp1 9289 . . . . . . . . . . 11 (((𝑁𝑟) ∈ ℕ0 ∧ (𝑁𝑝) ∈ ℕ0) → ((𝑁𝑟) ≤ (𝑁𝑝) ↔ (𝑁𝑟) < ((𝑁𝑝) + 1)))
4537, 43, 44syl2anc 411 . . . . . . . . . 10 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝑁𝑟) ≤ (𝑁𝑝) ↔ (𝑁𝑟) < ((𝑁𝑝) + 1)))
4642, 45mpbird 167 . . . . . . . . 9 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑟) ≤ (𝑁𝑝))
47 fznn0 10083 . . . . . . . . . 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 944 . . . . . . . 8 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑟) ∈ (0...(𝑁𝑝)))
5028, 29, 49rspcdva 2844 . . . . . . 7 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝐹𝑘) ≠ (𝐹‘(𝑁𝑟)))
5126adantr 276 . . . . . . . . 9 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑘) ∈ ω)
52 fvco3 5579 . . . . . . . . 9 ((𝑁:ω⟶ℕ0 ∧ (𝑁𝑘) ∈ ω) → ((𝐹𝑁)‘(𝑁𝑘)) = (𝐹‘(𝑁‘(𝑁𝑘))))
5316, 51, 52sylancr 414 . . . . . . . 8 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹𝑁)‘(𝑁𝑘)) = (𝐹‘(𝑁‘(𝑁𝑘))))
5425adantr 276 . . . . . . . . . 10 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 𝑘 ∈ ℕ0)
55 f1ocnvfv2 5769 . . . . . . . . . 10 ((𝑁:ω–1-1-onto→ℕ0𝑘 ∈ ℕ0) → (𝑁‘(𝑁𝑘)) = 𝑘)
564, 54, 55sylancr 414 . . . . . . . . 9 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘(𝑁𝑘)) = 𝑘)
5756fveq2d 5511 . . . . . . . 8 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝐹‘(𝑁‘(𝑁𝑘))) = (𝐹𝑘))
5853, 57eqtrd 2208 . . . . . . 7 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹𝑁)‘(𝑁𝑘)) = (𝐹𝑘))
59 fvco3 5579 . . . . . . . 8 ((𝑁:ω⟶ℕ0𝑟 ∈ ω) → ((𝐹𝑁)‘𝑟) = (𝐹‘(𝑁𝑟)))
6016, 35, 59sylancr 414 . . . . . . 7 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹𝑁)‘𝑟) = (𝐹‘(𝑁𝑟)))
6150, 58, 603netr4d 2378 . . . . . 6 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟))
6261ralrimiva 2548 . . . . 5 (((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) → ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟))
63 fveq2 5507 . . . . . . . 8 (𝑞 = (𝑁𝑘) → ((𝐹𝑁)‘𝑞) = ((𝐹𝑁)‘(𝑁𝑘)))
6463neeq1d 2363 . . . . . . 7 (𝑞 = (𝑁𝑘) → (((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟) ↔ ((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟)))
6564ralbidv 2475 . . . . . 6 (𝑞 = (𝑁𝑘) → (∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟) ↔ ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟)))
6665rspcev 2839 . . . . 5 (((𝑁𝑘) ∈ ω ∧ ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟)) → ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟))
6726, 62, 66syl2anc 411 . . . 4 (((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) → ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟))
6820, 67rexlimddv 2597 . . 3 ((𝜑𝑝 ∈ ω) → ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟))
6968ralrimiva 2548 . 2 (𝜑 → ∀𝑝 ∈ ω ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟))
70 id 19 . . . 4 (𝑎 = 𝑥𝑎 = 𝑥)
71 dmeq 4820 . . . . . . 7 (𝑎 = 𝑥 → dom 𝑎 = dom 𝑥)
7271opeq1d 3780 . . . . . 6 (𝑎 = 𝑥 → ⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩ = ⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩)
7372sneqd 3602 . . . . 5 (𝑎 = 𝑥 → {⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩} = {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩})
7470, 73uneq12d 3288 . . . 4 (𝑎 = 𝑥 → (𝑎 ∪ {⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩}) = (𝑥 ∪ {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩}))
7570, 74ifeq12d 3551 . . 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 5507 . . . . 5 (𝑏 = 𝑦 → ((𝐹𝑁)‘𝑏) = ((𝐹𝑁)‘𝑦))
77 imaeq2 4959 . . . . 5 (𝑏 = 𝑦 → ((𝐹𝑁) “ 𝑏) = ((𝐹𝑁) “ 𝑦))
7876, 77eleq12d 2246 . . . 4 (𝑏 = 𝑦 → (((𝐹𝑁)‘𝑏) ∈ ((𝐹𝑁) “ 𝑏) ↔ ((𝐹𝑁)‘𝑦) ∈ ((𝐹𝑁) “ 𝑦)))
7976opeq2d 3781 . . . . . 6 (𝑏 = 𝑦 → ⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩ = ⟨dom 𝑥, ((𝐹𝑁)‘𝑦)⟩)
8079sneqd 3602 . . . . 5 (𝑏 = 𝑦 → {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩} = {⟨dom 𝑥, ((𝐹𝑁)‘𝑦)⟩})
8180uneq2d 3287 . . . 4 (𝑏 = 𝑦 → (𝑥 ∪ {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩}) = (𝑥 ∪ {⟨dom 𝑥, ((𝐹𝑁)‘𝑦)⟩}))
8278, 81ifbieq2d 3556 . . 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 5945 . 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 2182 . . . 4 (𝑎 = 𝑥 → (𝑎 = 0 ↔ 𝑥 = 0))
85 fvoveq1 5888 . . . 4 (𝑎 = 𝑥 → (𝑁‘(𝑎 − 1)) = (𝑁‘(𝑥 − 1)))
8684, 85ifbieq2d 3556 . . 3 (𝑎 = 𝑥 → if(𝑎 = 0, ∅, (𝑁‘(𝑎 − 1))) = if(𝑥 = 0, ∅, (𝑁‘(𝑥 − 1))))
8786cbvmptv 4094 . 2 (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (𝑁‘(𝑎 − 1)))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (𝑁‘(𝑥 − 1))))
88 eqid 2175 . 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 5507 . . 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 3921 . 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 12389 1 (𝜑𝐴 ≈ ℕ)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  DECID wdc 834   = wceq 1353  wcel 2146  wne 2345  wral 2453  wrex 2454  cun 3125  c0 3420  ifcif 3532  {csn 3589  cop 3592   ciun 3882   class class class wbr 3998  cmpt 4059  suc csuc 4359  ωcom 4583  ccnv 4619  dom cdm 4620  cima 4623  ccom 4624  wf 5204  ontowfo 5206  1-1-ontowf1o 5207  cfv 5208  (class class class)co 5865  cmpo 5867  freccfrec 6381  pm cpm 6639  cen 6728  0cc0 7786  1c1 7787   + caddc 7789   < clt 7966  cle 7967  cmin 8102  cn 8892  0cn0 9149  cz 9226  ...cfz 9979  seqcseq 10415
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-ltadd 7902
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-er 6525  df-pm 6641  df-en 6731  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-inn 8893  df-n0 9150  df-z 9227  df-uz 9502  df-fz 9980  df-seqfrec 10416
This theorem is referenced by:  ennnfonelemr  12391
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