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Theorem ennnfonelemnn0 12472
Description: Lemma for ennnfone 12475. A version of ennnfonelemen 12471 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 10458 . . . . 5 𝑁:ω–1-1-onto→ℕ0
5 f1ofo 5487 . . . . 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 5467 . . 3 ((𝐹:ℕ0onto𝐴𝑁:ω–onto→ℕ0) → (𝐹𝑁):ω–onto𝐴)
92, 7, 8syl2anc 411 . 2 (𝜑 → (𝐹𝑁):ω–onto𝐴)
10 oveq2 5903 . . . . . . 7 (𝑛 = (𝑁𝑝) → (0...𝑛) = (0...(𝑁𝑝)))
1110raleqdv 2692 . . . . . 6 (𝑛 = (𝑁𝑝) → (∀𝑗 ∈ (0...𝑛)(𝐹𝑘) ≠ (𝐹𝑗) ↔ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗)))
1211rexbidv 2491 . . . . 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 5480 . . . . . . . 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 5672 . . . . 5 ((𝜑𝑝 ∈ ω) → (𝑁𝑝) ∈ ℕ0)
2012, 14, 19rspcdva 2861 . . . 4 ((𝜑𝑝 ∈ ω) → ∃𝑘 ∈ ℕ0𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))
21 f1ocnv 5493 . . . . . . . 8 (𝑁:ω–1-1-onto→ℕ0𝑁:ℕ01-1-onto→ω)
22 f1of 5480 . . . . . . . 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 5672 . . . . 5 (((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) → (𝑁𝑘) ∈ ω)
27 fveq2 5534 . . . . . . . . 9 (𝑗 = (𝑁𝑟) → (𝐹𝑗) = (𝐹‘(𝑁𝑟)))
2827neeq2d 2379 . . . . . . . 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 4612 . . . . . . . . . . . 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 4623 . . . . . . . . . . 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 5670 . . . . . . . . . 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 9294 . . . . . . . . . . . . 13 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → 0 ∈ ℤ)
3938, 3, 35, 33frec2uzltd 10433 . . . . . . . . . . . 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 10431 . . . . . . . . . . 11 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁‘suc 𝑝) = ((𝑁𝑝) + 1))
4240, 41breqtrd 4044 . . . . . . . . . 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 9345 . . . . . . . . . . 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 10142 . . . . . . . . . 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 946 . . . . . . . 8 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝑁𝑟) ∈ (0...(𝑁𝑝)))
5028, 29, 49rspcdva 2861 . . . . . . 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 5607 . . . . . . . . 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 5799 . . . . . . . . . 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 5538 . . . . . . . 8 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → (𝐹‘(𝑁‘(𝑁𝑘))) = (𝐹𝑘))
5853, 57eqtrd 2222 . . . . . . 7 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹𝑁)‘(𝑁𝑘)) = (𝐹𝑘))
59 fvco3 5607 . . . . . . . 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 2393 . . . . . 6 ((((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) ∧ 𝑟 ∈ suc 𝑝) → ((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟))
6261ralrimiva 2563 . . . . 5 (((𝜑𝑝 ∈ ω) ∧ (𝑘 ∈ ℕ0 ∧ ∀𝑗 ∈ (0...(𝑁𝑝))(𝐹𝑘) ≠ (𝐹𝑗))) → ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟))
63 fveq2 5534 . . . . . . . 8 (𝑞 = (𝑁𝑘) → ((𝐹𝑁)‘𝑞) = ((𝐹𝑁)‘(𝑁𝑘)))
6463neeq1d 2378 . . . . . . 7 (𝑞 = (𝑁𝑘) → (((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟) ↔ ((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟)))
6564ralbidv 2490 . . . . . 6 (𝑞 = (𝑁𝑘) → (∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟) ↔ ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘(𝑁𝑘)) ≠ ((𝐹𝑁)‘𝑟)))
6665rspcev 2856 . . . . 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 2612 . . 3 ((𝜑𝑝 ∈ ω) → ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟))
6968ralrimiva 2563 . 2 (𝜑 → ∀𝑝 ∈ ω ∃𝑞 ∈ ω ∀𝑟 ∈ suc 𝑝((𝐹𝑁)‘𝑞) ≠ ((𝐹𝑁)‘𝑟))
70 id 19 . . . 4 (𝑎 = 𝑥𝑎 = 𝑥)
71 dmeq 4845 . . . . . . 7 (𝑎 = 𝑥 → dom 𝑎 = dom 𝑥)
7271opeq1d 3799 . . . . . 6 (𝑎 = 𝑥 → ⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩ = ⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩)
7372sneqd 3620 . . . . 5 (𝑎 = 𝑥 → {⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩} = {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩})
7470, 73uneq12d 3305 . . . 4 (𝑎 = 𝑥 → (𝑎 ∪ {⟨dom 𝑎, ((𝐹𝑁)‘𝑏)⟩}) = (𝑥 ∪ {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩}))
7570, 74ifeq12d 3568 . . 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 5534 . . . . 5 (𝑏 = 𝑦 → ((𝐹𝑁)‘𝑏) = ((𝐹𝑁)‘𝑦))
77 imaeq2 4984 . . . . 5 (𝑏 = 𝑦 → ((𝐹𝑁) “ 𝑏) = ((𝐹𝑁) “ 𝑦))
7876, 77eleq12d 2260 . . . 4 (𝑏 = 𝑦 → (((𝐹𝑁)‘𝑏) ∈ ((𝐹𝑁) “ 𝑏) ↔ ((𝐹𝑁)‘𝑦) ∈ ((𝐹𝑁) “ 𝑦)))
7976opeq2d 3800 . . . . . 6 (𝑏 = 𝑦 → ⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩ = ⟨dom 𝑥, ((𝐹𝑁)‘𝑦)⟩)
8079sneqd 3620 . . . . 5 (𝑏 = 𝑦 → {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩} = {⟨dom 𝑥, ((𝐹𝑁)‘𝑦)⟩})
8180uneq2d 3304 . . . 4 (𝑏 = 𝑦 → (𝑥 ∪ {⟨dom 𝑥, ((𝐹𝑁)‘𝑏)⟩}) = (𝑥 ∪ {⟨dom 𝑥, ((𝐹𝑁)‘𝑦)⟩}))
8278, 81ifbieq2d 3573 . . 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 5975 . 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 2196 . . . 4 (𝑎 = 𝑥 → (𝑎 = 0 ↔ 𝑥 = 0))
85 fvoveq1 5918 . . . 4 (𝑎 = 𝑥 → (𝑁‘(𝑎 − 1)) = (𝑁‘(𝑥 − 1)))
8684, 85ifbieq2d 3573 . . 3 (𝑎 = 𝑥 → if(𝑎 = 0, ∅, (𝑁‘(𝑎 − 1))) = if(𝑥 = 0, ∅, (𝑁‘(𝑥 − 1))))
8786cbvmptv 4114 . 2 (𝑎 ∈ ℕ0 ↦ if(𝑎 = 0, ∅, (𝑁‘(𝑎 − 1)))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (𝑁‘(𝑥 − 1))))
88 eqid 2189 . 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 5534 . . 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 3940 . 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 12471 1 (𝜑𝐴 ≈ ℕ)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  DECID wdc 835   = wceq 1364  wcel 2160  wne 2360  wral 2468  wrex 2469  cun 3142  c0 3437  ifcif 3549  {csn 3607  cop 3610   ciun 3901   class class class wbr 4018  cmpt 4079  suc csuc 4383  ωcom 4607  ccnv 4643  dom cdm 4644  cima 4647  ccom 4648  wf 5231  ontowfo 5233  1-1-ontowf1o 5234  cfv 5235  (class class class)co 5895  cmpo 5897  freccfrec 6414  pm cpm 6674  cen 6763  0cc0 7840  1c1 7841   + caddc 7843   < clt 8021  cle 8022  cmin 8157  cn 8948  0cn0 9205  cz 9282  ...cfz 10037  seqcseq 10475
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 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-0id 7948  ax-rnegex 7949  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-ltadd 7956
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-frec 6415  df-er 6558  df-pm 6676  df-en 6766  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-inn 8949  df-n0 9206  df-z 9283  df-uz 9558  df-fz 10038  df-seqfrec 10476
This theorem is referenced by:  ennnfonelemr  12473
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