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Theorem ennnfonelemkh 11770
Description: Lemma for ennnfone 11783. 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 5375 . . . . . . 7 (𝑤 = 0 → (𝐻𝑤) = (𝐻‘0))
32dmeqd 4701 . . . . . 6 (𝑤 = 0 → dom (𝐻𝑤) = dom (𝐻‘0))
4 fveq2 5375 . . . . . 6 (𝑤 = 0 → (𝑁𝑤) = (𝑁‘0))
53, 4sseq12d 3094 . . . . 5 (𝑤 = 0 → (dom (𝐻𝑤) ⊆ (𝑁𝑤) ↔ dom (𝐻‘0) ⊆ (𝑁‘0)))
65imbi2d 229 . . . 4 (𝑤 = 0 → ((𝜑 → dom (𝐻𝑤) ⊆ (𝑁𝑤)) ↔ (𝜑 → dom (𝐻‘0) ⊆ (𝑁‘0))))
7 fveq2 5375 . . . . . . 7 (𝑤 = 𝑚 → (𝐻𝑤) = (𝐻𝑚))
87dmeqd 4701 . . . . . 6 (𝑤 = 𝑚 → dom (𝐻𝑤) = dom (𝐻𝑚))
9 fveq2 5375 . . . . . 6 (𝑤 = 𝑚 → (𝑁𝑤) = (𝑁𝑚))
108, 9sseq12d 3094 . . . . 5 (𝑤 = 𝑚 → (dom (𝐻𝑤) ⊆ (𝑁𝑤) ↔ dom (𝐻𝑚) ⊆ (𝑁𝑚)))
1110imbi2d 229 . . . 4 (𝑤 = 𝑚 → ((𝜑 → dom (𝐻𝑤) ⊆ (𝑁𝑤)) ↔ (𝜑 → dom (𝐻𝑚) ⊆ (𝑁𝑚))))
12 fveq2 5375 . . . . . . 7 (𝑤 = (𝑚 + 1) → (𝐻𝑤) = (𝐻‘(𝑚 + 1)))
1312dmeqd 4701 . . . . . 6 (𝑤 = (𝑚 + 1) → dom (𝐻𝑤) = dom (𝐻‘(𝑚 + 1)))
14 fveq2 5375 . . . . . 6 (𝑤 = (𝑚 + 1) → (𝑁𝑤) = (𝑁‘(𝑚 + 1)))
1513, 14sseq12d 3094 . . . . 5 (𝑤 = (𝑚 + 1) → (dom (𝐻𝑤) ⊆ (𝑁𝑤) ↔ dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1))))
1615imbi2d 229 . . . 4 (𝑤 = (𝑚 + 1) → ((𝜑 → dom (𝐻𝑤) ⊆ (𝑁𝑤)) ↔ (𝜑 → dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1)))))
17 fveq2 5375 . . . . . . 7 (𝑤 = 𝑃 → (𝐻𝑤) = (𝐻𝑃))
1817dmeqd 4701 . . . . . 6 (𝑤 = 𝑃 → dom (𝐻𝑤) = dom (𝐻𝑃))
19 fveq2 5375 . . . . . 6 (𝑤 = 𝑃 → (𝑁𝑤) = (𝑁𝑃))
2018, 19sseq12d 3094 . . . . 5 (𝑤 = 𝑃 → (dom (𝐻𝑤) ⊆ (𝑁𝑤) ↔ dom (𝐻𝑃) ⊆ (𝑁𝑃)))
2120imbi2d 229 . . . 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 11763 . . . . . . . 8 (𝜑 → (𝐻‘0) = ∅)
3029dmeqd 4701 . . . . . . 7 (𝜑 → dom (𝐻‘0) = dom ∅)
31 dm0 4713 . . . . . . 7 dom ∅ = ∅
3230, 31syl6eq 2163 . . . . . 6 (𝜑 → dom (𝐻‘0) = ∅)
33 0ss 3367 . . . . . 6 ∅ ⊆ (𝑁‘0)
3432, 33syl6eqss 3115 . . . . 5 (𝜑 → dom (𝐻‘0) ⊆ (𝑁‘0))
3534a1i 9 . . . 4 (0 ∈ ℤ → (𝜑 → dom (𝐻‘0) ⊆ (𝑁‘0)))
3626frechashgf1o 10094 . . . . . . . . . . . . . 14 𝑁:ω–1-1-onto→ℕ0
37 f1of 5323 . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
4023ad2antrr 477 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 𝐹:ω–onto𝐴)
4124ad2antrr 477 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹𝑘) ≠ (𝐹𝑗))
42 simplr 502 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 𝑚 ∈ (ℤ‘0))
43 nn0uz 9262 . . . . . . . . . . . . . . . . 17 0 = (ℤ‘0)
4442, 43syl6eleqr 2208 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 𝑚 ∈ ℕ0)
45 peano2nn0 8921 . . . . . . . . . . . . . . . 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 11766 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → dom (𝐻‘(𝑚 + 1)) ∈ ω)
4847adantr 272 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) ∈ ω)
4938, 48ffvelrnd 5510 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ∈ ℕ0)
5049nn0red 8935 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ∈ ℝ)
5144nn0red 8935 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 𝑚 ∈ ℝ)
5251adantr 272 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → 𝑚 ∈ ℝ)
53 peano2re 7821 . . . . . . . . . . . 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 11764 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝐻‘(𝑚 + 1)) = if((𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)), (𝐻𝑚), ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩})))
5655adantr 272 . . . . . . . . . . . . . . 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 109 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)))
5857iftrued 3447 . . . . . . . . . . . . . . 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 2147 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝐻‘(𝑚 + 1)) = (𝐻𝑚))
6059dmeqd 4701 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) = dom (𝐻𝑚))
6160fveq2d 5379 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) = (𝑁‘dom (𝐻𝑚)))
62 simpr 109 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → dom (𝐻𝑚) ⊆ (𝑁𝑚))
63 0zd 8970 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 0 ∈ ℤ)
6439, 40, 41, 25, 26, 27, 28, 44ennnfonelemom 11766 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → dom (𝐻𝑚) ∈ ω)
65 f1ocnv 5336 . . . . . . . . . . . . . . . . . . . 20 (𝑁:ω–1-1-onto→ℕ0𝑁:ℕ01-1-onto→ω)
6636, 65ax-mp 7 . . . . . . . . . . . . . . . . . . 19 𝑁:ℕ01-1-onto→ω
67 f1of 5323 . . . . . . . . . . . . . . . . . . 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, 69ffvelrnd 5510 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ0 → (𝑁𝑚) ∈ ω)
7144, 70syl 14 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁𝑚) ∈ ω)
7263, 26, 64, 71frec2uzled 10095 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (dom (𝐻𝑚) ⊆ (𝑁𝑚) ↔ (𝑁‘dom (𝐻𝑚)) ≤ (𝑁‘(𝑁𝑚))))
7362, 72mpbid 146 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁‘dom (𝐻𝑚)) ≤ (𝑁‘(𝑁𝑚)))
74 f1ocnvfv2 5633 . . . . . . . . . . . . . . 15 ((𝑁:ω–1-1-onto→ℕ0𝑚 ∈ ℕ0) → (𝑁‘(𝑁𝑚)) = 𝑚)
7536, 44, 74sylancr 408 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁‘(𝑁𝑚)) = 𝑚)
7673, 75breqtrd 3919 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁‘dom (𝐻𝑚)) ≤ 𝑚)
7776adantr 272 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻𝑚)) ≤ 𝑚)
7861, 77eqbrtrd 3915 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ 𝑚)
7952lep1d 8599 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → 𝑚 ≤ (𝑚 + 1))
8050, 52, 54, 78, 79letrd 7809 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑚 + 1))
81 f1ocnvfv2 5633 . . . . . . . . . . . 12 ((𝑁:ω–1-1-onto→ℕ0 ∧ (𝑚 + 1) ∈ ℕ0) → (𝑁‘(𝑁‘(𝑚 + 1))) = (𝑚 + 1))
8236, 46, 81sylancr 408 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁‘(𝑁‘(𝑚 + 1))) = (𝑚 + 1))
8382adantr 272 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘(𝑁‘(𝑚 + 1))) = (𝑚 + 1))
8480, 83breqtrrd 3921 . . . . . . . . 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, 46ffvelrnd 5510 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁‘(𝑚 + 1)) ∈ ω)
8763, 26, 47, 86frec2uzled 10095 . . . . . . . . . 10 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1)) ↔ (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑁‘(𝑁‘(𝑚 + 1)))))
8887adantr 272 . . . . . . . . 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 166 . . . . . . . 8 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1)))
9055adantr 272 . . . . . . . . . . . . . . . . . 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 109 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)))
9291iffalsed 3450 . . . . . . . . . . . . . . . . . 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 2147 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝐻‘(𝑚 + 1)) = ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩}))
9493dmeqd 4701 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) = dom ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩}))
95 dmun 4706 . . . . . . . . . . . . . . . 16 dom ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩}) = (dom (𝐻𝑚) ∪ dom {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩})
9694, 95syl6eq 2163 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) = (dom (𝐻𝑚) ∪ dom {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩}))
97 fof 5303 . . . . . . . . . . . . . . . . . . . 20 (𝐹:ω–onto𝐴𝐹:ω⟶𝐴)
9840, 97syl 14 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 𝐹:ω⟶𝐴)
9998, 71ffvelrnd 5510 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝐹‘(𝑁𝑚)) ∈ 𝐴)
10099adantr 272 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝐹‘(𝑁𝑚)) ∈ 𝐴)
101 dmsnopg 4968 . . . . . . . . . . . . . . . . 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 3196 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (dom (𝐻𝑚) ∪ dom {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩}) = (dom (𝐻𝑚) ∪ {dom (𝐻𝑚)}))
10496, 103eqtrd 2147 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) = (dom (𝐻𝑚) ∪ {dom (𝐻𝑚)}))
105 df-suc 4253 . . . . . . . . . . . . . 14 suc dom (𝐻𝑚) = (dom (𝐻𝑚) ∪ {dom (𝐻𝑚)})
106104, 105syl6eqr 2165 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) = suc dom (𝐻𝑚))
107 simplr 502 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻𝑚) ⊆ (𝑁𝑚))
10871adantr 272 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁𝑚) ∈ ω)
109 nnsucsssuc 6342 . . . . . . . . . . . . . . 15 ((dom (𝐻𝑚) ∈ ω ∧ (𝑁𝑚) ∈ ω) → (dom (𝐻𝑚) ⊆ (𝑁𝑚) ↔ suc dom (𝐻𝑚) ⊆ suc (𝑁𝑚)))
11064, 108, 109syl2an2r 567 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (dom (𝐻𝑚) ⊆ (𝑁𝑚) ↔ suc dom (𝐻𝑚) ⊆ suc (𝑁𝑚)))
111107, 110mpbid 146 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → suc dom (𝐻𝑚) ⊆ suc (𝑁𝑚))
112106, 111eqsstrd 3099 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) ⊆ suc (𝑁𝑚))
113 0zd 8970 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → 0 ∈ ℤ)
11447adantr 272 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) ∈ ω)
115 peano2 4469 . . . . . . . . . . . . . 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 10095 . . . . . . . . . . . 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 146 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑁‘suc (𝑁𝑚)))
119113, 26, 108frec2uzsucd 10067 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘suc (𝑁𝑚)) = ((𝑁‘(𝑁𝑚)) + 1))
12075adantr 272 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘(𝑁𝑚)) = 𝑚)
121120oveq1d 5743 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → ((𝑁‘(𝑁𝑚)) + 1) = (𝑚 + 1))
122119, 121eqtrd 2147 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘suc (𝑁𝑚)) = (𝑚 + 1))
123118, 122breqtrd 3919 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑚 + 1))
12482adantr 272 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘(𝑁‘(𝑚 + 1))) = (𝑚 + 1))
125123, 124breqtrrd 3921 . . . . . . . . 9 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑁‘(𝑁‘(𝑚 + 1))))
12686adantr 272 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘(𝑚 + 1)) ∈ ω)
127113, 26, 114, 126frec2uzled 10095 . . . . . . . . 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 166 . . . . . . . 8 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1)))
12939, 40, 71ennnfonelemdc 11757 . . . . . . . . 9 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → DECID (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)))
130 exmiddc 804 . . . . . . . . 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 770 . . . . . . 7 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1)))
133132ex 114 . . . . . 6 ((𝜑𝑚 ∈ (ℤ‘0)) → (dom (𝐻𝑚) ⊆ (𝑁𝑚) → dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1))))
134133expcom 115 . . . . 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 9285 . . 3 (𝑃 ∈ (ℤ‘0) → (𝜑 → dom (𝐻𝑃) ⊆ (𝑁𝑃)))
137136, 43eleq2s 2209 . 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 103  wb 104  wo 680  DECID wdc 802   = wceq 1314  wcel 1463  wne 2282  wral 2390  wrex 2391  cun 3035  wss 3037  c0 3329  ifcif 3440  {csn 3493  cop 3496   class class class wbr 3895  cmpt 3949  suc csuc 4247  ωcom 4464  ccnv 4498  dom cdm 4499  cima 4502  wf 5077  ontowfo 5079  1-1-ontowf1o 5080  cfv 5081  (class class class)co 5728  cmpo 5730  freccfrec 6241  pm cpm 6497  cr 7546  0cc0 7547  1c1 7548   + caddc 7550  cle 7725  cmin 7856  0cn0 8881  cz 8958  cuz 9228  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-pm 6499  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-seqfrec 10112
This theorem is referenced by: (None)
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