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Theorem ennnfonelemkh 12416
Description: Lemma for ennnfone 12429. Because we add zero or one entries for each new index, the length of each sequence is no greater than its index. (Contributed by Jim Kingdon, 19-Jul-2023.)
Hypotheses
Ref Expression
ennnfonelemh.dceq (𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
ennnfonelemh.f (𝜑𝐹:ω–onto𝐴)
ennnfonelemh.ne (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹𝑘) ≠ (𝐹𝑗))
ennnfonelemh.g 𝐺 = (𝑥 ∈ (𝐴pm ω), 𝑦 ∈ ω ↦ if((𝐹𝑦) ∈ (𝐹𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹𝑦)⟩})))
ennnfonelemh.n 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
ennnfonelemh.j 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (𝑁‘(𝑥 − 1))))
ennnfonelemh.h 𝐻 = seq0(𝐺, 𝐽)
ennnfonelemkh.p (𝜑𝑃 ∈ ℕ0)
Assertion
Ref Expression
ennnfonelemkh (𝜑 → dom (𝐻𝑃) ⊆ (𝑁𝑃))
Distinct variable groups:   𝐴,𝑗,𝑥,𝑦   𝑥,𝐹,𝑦   𝑗,𝐺   𝑗,𝐻,𝑥,𝑦   𝑗,𝐽   𝑗,𝑁,𝑥,𝑦   𝜑,𝑗,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑘,𝑛)   𝐴(𝑘,𝑛)   𝑃(𝑥,𝑦,𝑗,𝑘,𝑛)   𝐹(𝑗,𝑘,𝑛)   𝐺(𝑥,𝑦,𝑘,𝑛)   𝐻(𝑘,𝑛)   𝐽(𝑥,𝑦,𝑘,𝑛)   𝑁(𝑘,𝑛)

Proof of Theorem ennnfonelemkh
Dummy variables 𝑚 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ennnfonelemkh.p . 2 (𝜑𝑃 ∈ ℕ0)
2 fveq2 5517 . . . . . . 7 (𝑤 = 0 → (𝐻𝑤) = (𝐻‘0))
32dmeqd 4831 . . . . . 6 (𝑤 = 0 → dom (𝐻𝑤) = dom (𝐻‘0))
4 fveq2 5517 . . . . . 6 (𝑤 = 0 → (𝑁𝑤) = (𝑁‘0))
53, 4sseq12d 3188 . . . . 5 (𝑤 = 0 → (dom (𝐻𝑤) ⊆ (𝑁𝑤) ↔ dom (𝐻‘0) ⊆ (𝑁‘0)))
65imbi2d 230 . . . 4 (𝑤 = 0 → ((𝜑 → dom (𝐻𝑤) ⊆ (𝑁𝑤)) ↔ (𝜑 → dom (𝐻‘0) ⊆ (𝑁‘0))))
7 fveq2 5517 . . . . . . 7 (𝑤 = 𝑚 → (𝐻𝑤) = (𝐻𝑚))
87dmeqd 4831 . . . . . 6 (𝑤 = 𝑚 → dom (𝐻𝑤) = dom (𝐻𝑚))
9 fveq2 5517 . . . . . 6 (𝑤 = 𝑚 → (𝑁𝑤) = (𝑁𝑚))
108, 9sseq12d 3188 . . . . 5 (𝑤 = 𝑚 → (dom (𝐻𝑤) ⊆ (𝑁𝑤) ↔ dom (𝐻𝑚) ⊆ (𝑁𝑚)))
1110imbi2d 230 . . . 4 (𝑤 = 𝑚 → ((𝜑 → dom (𝐻𝑤) ⊆ (𝑁𝑤)) ↔ (𝜑 → dom (𝐻𝑚) ⊆ (𝑁𝑚))))
12 fveq2 5517 . . . . . . 7 (𝑤 = (𝑚 + 1) → (𝐻𝑤) = (𝐻‘(𝑚 + 1)))
1312dmeqd 4831 . . . . . 6 (𝑤 = (𝑚 + 1) → dom (𝐻𝑤) = dom (𝐻‘(𝑚 + 1)))
14 fveq2 5517 . . . . . 6 (𝑤 = (𝑚 + 1) → (𝑁𝑤) = (𝑁‘(𝑚 + 1)))
1513, 14sseq12d 3188 . . . . 5 (𝑤 = (𝑚 + 1) → (dom (𝐻𝑤) ⊆ (𝑁𝑤) ↔ dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1))))
1615imbi2d 230 . . . 4 (𝑤 = (𝑚 + 1) → ((𝜑 → dom (𝐻𝑤) ⊆ (𝑁𝑤)) ↔ (𝜑 → dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1)))))
17 fveq2 5517 . . . . . . 7 (𝑤 = 𝑃 → (𝐻𝑤) = (𝐻𝑃))
1817dmeqd 4831 . . . . . 6 (𝑤 = 𝑃 → dom (𝐻𝑤) = dom (𝐻𝑃))
19 fveq2 5517 . . . . . 6 (𝑤 = 𝑃 → (𝑁𝑤) = (𝑁𝑃))
2018, 19sseq12d 3188 . . . . 5 (𝑤 = 𝑃 → (dom (𝐻𝑤) ⊆ (𝑁𝑤) ↔ dom (𝐻𝑃) ⊆ (𝑁𝑃)))
2120imbi2d 230 . . . 4 (𝑤 = 𝑃 → ((𝜑 → dom (𝐻𝑤) ⊆ (𝑁𝑤)) ↔ (𝜑 → dom (𝐻𝑃) ⊆ (𝑁𝑃))))
22 ennnfonelemh.dceq . . . . . . . . 9 (𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
23 ennnfonelemh.f . . . . . . . . 9 (𝜑𝐹:ω–onto𝐴)
24 ennnfonelemh.ne . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹𝑘) ≠ (𝐹𝑗))
25 ennnfonelemh.g . . . . . . . . 9 𝐺 = (𝑥 ∈ (𝐴pm ω), 𝑦 ∈ ω ↦ if((𝐹𝑦) ∈ (𝐹𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹𝑦)⟩})))
26 ennnfonelemh.n . . . . . . . . 9 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
27 ennnfonelemh.j . . . . . . . . 9 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (𝑁‘(𝑥 − 1))))
28 ennnfonelemh.h . . . . . . . . 9 𝐻 = seq0(𝐺, 𝐽)
2922, 23, 24, 25, 26, 27, 28ennnfonelem0 12409 . . . . . . . 8 (𝜑 → (𝐻‘0) = ∅)
3029dmeqd 4831 . . . . . . 7 (𝜑 → dom (𝐻‘0) = dom ∅)
31 dm0 4843 . . . . . . 7 dom ∅ = ∅
3230, 31eqtrdi 2226 . . . . . 6 (𝜑 → dom (𝐻‘0) = ∅)
33 0ss 3463 . . . . . 6 ∅ ⊆ (𝑁‘0)
3432, 33eqsstrdi 3209 . . . . 5 (𝜑 → dom (𝐻‘0) ⊆ (𝑁‘0))
3534a1i 9 . . . 4 (0 ∈ ℤ → (𝜑 → dom (𝐻‘0) ⊆ (𝑁‘0)))
3626frechashgf1o 10431 . . . . . . . . . . . . . 14 𝑁:ω–1-1-onto→ℕ0
37 f1of 5463 . . . . . . . . . . . . . 14 (𝑁:ω–1-1-onto→ℕ0𝑁:ω⟶ℕ0)
3836, 37mp1i 10 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → 𝑁:ω⟶ℕ0)
3922ad2antrr 488 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
4023ad2antrr 488 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 𝐹:ω–onto𝐴)
4124ad2antrr 488 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹𝑘) ≠ (𝐹𝑗))
42 simplr 528 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 𝑚 ∈ (ℤ‘0))
43 nn0uz 9565 . . . . . . . . . . . . . . . . 17 0 = (ℤ‘0)
4442, 43eleqtrrdi 2271 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 𝑚 ∈ ℕ0)
45 peano2nn0 9219 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
4644, 45syl 14 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑚 + 1) ∈ ℕ0)
4739, 40, 41, 25, 26, 27, 28, 46ennnfonelemom 12412 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → dom (𝐻‘(𝑚 + 1)) ∈ ω)
4847adantr 276 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) ∈ ω)
4938, 48ffvelcdmd 5655 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ∈ ℕ0)
5049nn0red 9233 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ∈ ℝ)
5144nn0red 9233 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 𝑚 ∈ ℝ)
5251adantr 276 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → 𝑚 ∈ ℝ)
53 peano2re 8096 . . . . . . . . . . . 12 (𝑚 ∈ ℝ → (𝑚 + 1) ∈ ℝ)
5452, 53syl 14 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑚 + 1) ∈ ℝ)
5539, 40, 41, 25, 26, 27, 28, 44ennnfonelemp1 12410 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝐻‘(𝑚 + 1)) = if((𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)), (𝐻𝑚), ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩})))
5655adantr 276 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝐻‘(𝑚 + 1)) = if((𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)), (𝐻𝑚), ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩})))
57 simpr 110 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)))
5857iftrued 3543 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → if((𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)), (𝐻𝑚), ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩})) = (𝐻𝑚))
5956, 58eqtrd 2210 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝐻‘(𝑚 + 1)) = (𝐻𝑚))
6059dmeqd 4831 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) = dom (𝐻𝑚))
6160fveq2d 5521 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) = (𝑁‘dom (𝐻𝑚)))
62 simpr 110 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → dom (𝐻𝑚) ⊆ (𝑁𝑚))
63 0zd 9268 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 0 ∈ ℤ)
6439, 40, 41, 25, 26, 27, 28, 44ennnfonelemom 12412 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → dom (𝐻𝑚) ∈ ω)
65 f1ocnv 5476 . . . . . . . . . . . . . . . . . . . 20 (𝑁:ω–1-1-onto→ℕ0𝑁:ℕ01-1-onto→ω)
6636, 65ax-mp 5 . . . . . . . . . . . . . . . . . . 19 𝑁:ℕ01-1-onto→ω
67 f1of 5463 . . . . . . . . . . . . . . . . . . 19 (𝑁:ℕ01-1-onto→ω → 𝑁:ℕ0⟶ω)
6866, 67mp1i 10 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕ0𝑁:ℕ0⟶ω)
69 id 19 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕ0𝑚 ∈ ℕ0)
7068, 69ffvelcdmd 5655 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ0 → (𝑁𝑚) ∈ ω)
7144, 70syl 14 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁𝑚) ∈ ω)
7263, 26, 64, 71frec2uzled 10432 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (dom (𝐻𝑚) ⊆ (𝑁𝑚) ↔ (𝑁‘dom (𝐻𝑚)) ≤ (𝑁‘(𝑁𝑚))))
7362, 72mpbid 147 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁‘dom (𝐻𝑚)) ≤ (𝑁‘(𝑁𝑚)))
74 f1ocnvfv2 5782 . . . . . . . . . . . . . . 15 ((𝑁:ω–1-1-onto→ℕ0𝑚 ∈ ℕ0) → (𝑁‘(𝑁𝑚)) = 𝑚)
7536, 44, 74sylancr 414 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁‘(𝑁𝑚)) = 𝑚)
7673, 75breqtrd 4031 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁‘dom (𝐻𝑚)) ≤ 𝑚)
7776adantr 276 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻𝑚)) ≤ 𝑚)
7861, 77eqbrtrd 4027 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ 𝑚)
7952lep1d 8891 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → 𝑚 ≤ (𝑚 + 1))
8050, 52, 54, 78, 79letrd 8084 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑚 + 1))
81 f1ocnvfv2 5782 . . . . . . . . . . . 12 ((𝑁:ω–1-1-onto→ℕ0 ∧ (𝑚 + 1) ∈ ℕ0) → (𝑁‘(𝑁‘(𝑚 + 1))) = (𝑚 + 1))
8236, 46, 81sylancr 414 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁‘(𝑁‘(𝑚 + 1))) = (𝑚 + 1))
8382adantr 276 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘(𝑁‘(𝑚 + 1))) = (𝑚 + 1))
8480, 83breqtrrd 4033 . . . . . . . . 9 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑁‘(𝑁‘(𝑚 + 1))))
8566, 67mp1i 10 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 𝑁:ℕ0⟶ω)
8685, 46ffvelcdmd 5655 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁‘(𝑚 + 1)) ∈ ω)
8763, 26, 47, 86frec2uzled 10432 . . . . . . . . . 10 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1)) ↔ (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑁‘(𝑁‘(𝑚 + 1)))))
8887adantr 276 . . . . . . . . 9 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1)) ↔ (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑁‘(𝑁‘(𝑚 + 1)))))
8984, 88mpbird 167 . . . . . . . 8 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1)))
9055adantr 276 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝐻‘(𝑚 + 1)) = if((𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)), (𝐻𝑚), ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩})))
91 simpr 110 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)))
9291iffalsed 3546 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → if((𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)), (𝐻𝑚), ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩})) = ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩}))
9390, 92eqtrd 2210 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝐻‘(𝑚 + 1)) = ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩}))
9493dmeqd 4831 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) = dom ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩}))
95 dmun 4836 . . . . . . . . . . . . . . . 16 dom ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩}) = (dom (𝐻𝑚) ∪ dom {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩})
9694, 95eqtrdi 2226 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) = (dom (𝐻𝑚) ∪ dom {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩}))
97 fof 5440 . . . . . . . . . . . . . . . . . . . 20 (𝐹:ω–onto𝐴𝐹:ω⟶𝐴)
9840, 97syl 14 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 𝐹:ω⟶𝐴)
9998, 71ffvelcdmd 5655 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝐹‘(𝑁𝑚)) ∈ 𝐴)
10099adantr 276 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝐹‘(𝑁𝑚)) ∈ 𝐴)
101 dmsnopg 5102 . . . . . . . . . . . . . . . . 17 ((𝐹‘(𝑁𝑚)) ∈ 𝐴 → dom {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩} = {dom (𝐻𝑚)})
102100, 101syl 14 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩} = {dom (𝐻𝑚)})
103102uneq2d 3291 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (dom (𝐻𝑚) ∪ dom {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩}) = (dom (𝐻𝑚) ∪ {dom (𝐻𝑚)}))
10496, 103eqtrd 2210 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) = (dom (𝐻𝑚) ∪ {dom (𝐻𝑚)}))
105 df-suc 4373 . . . . . . . . . . . . . 14 suc dom (𝐻𝑚) = (dom (𝐻𝑚) ∪ {dom (𝐻𝑚)})
106104, 105eqtr4di 2228 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) = suc dom (𝐻𝑚))
107 simplr 528 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻𝑚) ⊆ (𝑁𝑚))
10871adantr 276 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁𝑚) ∈ ω)
109 nnsucsssuc 6496 . . . . . . . . . . . . . . 15 ((dom (𝐻𝑚) ∈ ω ∧ (𝑁𝑚) ∈ ω) → (dom (𝐻𝑚) ⊆ (𝑁𝑚) ↔ suc dom (𝐻𝑚) ⊆ suc (𝑁𝑚)))
11064, 108, 109syl2an2r 595 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (dom (𝐻𝑚) ⊆ (𝑁𝑚) ↔ suc dom (𝐻𝑚) ⊆ suc (𝑁𝑚)))
111107, 110mpbid 147 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → suc dom (𝐻𝑚) ⊆ suc (𝑁𝑚))
112106, 111eqsstrd 3193 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) ⊆ suc (𝑁𝑚))
113 0zd 9268 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → 0 ∈ ℤ)
11447adantr 276 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) ∈ ω)
115 peano2 4596 . . . . . . . . . . . . . 14 ((𝑁𝑚) ∈ ω → suc (𝑁𝑚) ∈ ω)
116108, 115syl 14 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → suc (𝑁𝑚) ∈ ω)
117113, 26, 114, 116frec2uzled 10432 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (dom (𝐻‘(𝑚 + 1)) ⊆ suc (𝑁𝑚) ↔ (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑁‘suc (𝑁𝑚))))
118112, 117mpbid 147 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑁‘suc (𝑁𝑚)))
119113, 26, 108frec2uzsucd 10404 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘suc (𝑁𝑚)) = ((𝑁‘(𝑁𝑚)) + 1))
12075adantr 276 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘(𝑁𝑚)) = 𝑚)
121120oveq1d 5893 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → ((𝑁‘(𝑁𝑚)) + 1) = (𝑚 + 1))
122119, 121eqtrd 2210 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘suc (𝑁𝑚)) = (𝑚 + 1))
123118, 122breqtrd 4031 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑚 + 1))
12482adantr 276 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘(𝑁‘(𝑚 + 1))) = (𝑚 + 1))
125123, 124breqtrrd 4033 . . . . . . . . 9 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑁‘(𝑁‘(𝑚 + 1))))
12686adantr 276 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘(𝑚 + 1)) ∈ ω)
127113, 26, 114, 126frec2uzled 10432 . . . . . . . . 9 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1)) ↔ (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑁‘(𝑁‘(𝑚 + 1)))))
128125, 127mpbird 167 . . . . . . . 8 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1)))
12939, 40, 71ennnfonelemdc 12403 . . . . . . . . 9 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → DECID (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)))
130 exmiddc 836 . . . . . . . . 9 (DECID (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)) → ((𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)) ∨ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))))
131129, 130syl 14 . . . . . . . 8 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → ((𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)) ∨ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))))
13289, 128, 131mpjaodan 798 . . . . . . 7 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1)))
133132ex 115 . . . . . 6 ((𝜑𝑚 ∈ (ℤ‘0)) → (dom (𝐻𝑚) ⊆ (𝑁𝑚) → dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1))))
134133expcom 116 . . . . 5 (𝑚 ∈ (ℤ‘0) → (𝜑 → (dom (𝐻𝑚) ⊆ (𝑁𝑚) → dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1)))))
135134a2d 26 . . . 4 (𝑚 ∈ (ℤ‘0) → ((𝜑 → dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝜑 → dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1)))))
1366, 11, 16, 21, 35, 135uzind4 9591 . . 3 (𝑃 ∈ (ℤ‘0) → (𝜑 → dom (𝐻𝑃) ⊆ (𝑁𝑃)))
137136, 43eleq2s 2272 . 2 (𝑃 ∈ ℕ0 → (𝜑 → dom (𝐻𝑃) ⊆ (𝑁𝑃)))
1381, 137mpcom 36 1 (𝜑 → dom (𝐻𝑃) ⊆ (𝑁𝑃))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834   = wceq 1353  wcel 2148  wne 2347  wral 2455  wrex 2456  cun 3129  wss 3131  c0 3424  ifcif 3536  {csn 3594  cop 3597   class class class wbr 4005  cmpt 4066  suc csuc 4367  ωcom 4591  ccnv 4627  dom cdm 4628  cima 4631  wf 5214  ontowfo 5216  1-1-ontowf1o 5217  cfv 5218  (class class class)co 5878  cmpo 5880  freccfrec 6394  pm cpm 6652  cr 7813  0cc0 7814  1c1 7815   + caddc 7817  cle 7996  cmin 8131  0cn0 9179  cz 9256  cuz 9531  seqcseq 10448
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-ltadd 7930
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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-pm 6654  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-inn 8923  df-n0 9180  df-z 9257  df-uz 9532  df-seqfrec 10449
This theorem is referenced by: (None)
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