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Theorem ennnfonelemkh 12124
 Description: Lemma for ennnfone 12137. 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 5467 . . . . . . 7 (𝑤 = 0 → (𝐻𝑤) = (𝐻‘0))
32dmeqd 4787 . . . . . 6 (𝑤 = 0 → dom (𝐻𝑤) = dom (𝐻‘0))
4 fveq2 5467 . . . . . 6 (𝑤 = 0 → (𝑁𝑤) = (𝑁‘0))
53, 4sseq12d 3159 . . . . 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 5467 . . . . . . 7 (𝑤 = 𝑚 → (𝐻𝑤) = (𝐻𝑚))
87dmeqd 4787 . . . . . 6 (𝑤 = 𝑚 → dom (𝐻𝑤) = dom (𝐻𝑚))
9 fveq2 5467 . . . . . 6 (𝑤 = 𝑚 → (𝑁𝑤) = (𝑁𝑚))
108, 9sseq12d 3159 . . . . 5 (𝑤 = 𝑚 → (dom (𝐻𝑤) ⊆ (𝑁𝑤) ↔ dom (𝐻𝑚) ⊆ (𝑁𝑚)))
1110imbi2d 229 . . . 4 (𝑤 = 𝑚 → ((𝜑 → dom (𝐻𝑤) ⊆ (𝑁𝑤)) ↔ (𝜑 → dom (𝐻𝑚) ⊆ (𝑁𝑚))))
12 fveq2 5467 . . . . . . 7 (𝑤 = (𝑚 + 1) → (𝐻𝑤) = (𝐻‘(𝑚 + 1)))
1312dmeqd 4787 . . . . . 6 (𝑤 = (𝑚 + 1) → dom (𝐻𝑤) = dom (𝐻‘(𝑚 + 1)))
14 fveq2 5467 . . . . . 6 (𝑤 = (𝑚 + 1) → (𝑁𝑤) = (𝑁‘(𝑚 + 1)))
1513, 14sseq12d 3159 . . . . 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 5467 . . . . . . 7 (𝑤 = 𝑃 → (𝐻𝑤) = (𝐻𝑃))
1817dmeqd 4787 . . . . . 6 (𝑤 = 𝑃 → dom (𝐻𝑤) = dom (𝐻𝑃))
19 fveq2 5467 . . . . . 6 (𝑤 = 𝑃 → (𝑁𝑤) = (𝑁𝑃))
2018, 19sseq12d 3159 . . . . 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 12117 . . . . . . . 8 (𝜑 → (𝐻‘0) = ∅)
3029dmeqd 4787 . . . . . . 7 (𝜑 → dom (𝐻‘0) = dom ∅)
31 dm0 4799 . . . . . . 7 dom ∅ = ∅
3230, 31eqtrdi 2206 . . . . . 6 (𝜑 → dom (𝐻‘0) = ∅)
33 0ss 3432 . . . . . 6 ∅ ⊆ (𝑁‘0)
3432, 33eqsstrdi 3180 . . . . 5 (𝜑 → dom (𝐻‘0) ⊆ (𝑁‘0))
3534a1i 9 . . . 4 (0 ∈ ℤ → (𝜑 → dom (𝐻‘0) ⊆ (𝑁‘0)))
3626frechashgf1o 10320 . . . . . . . . . . . . . 14 𝑁:ω–1-1-onto→ℕ0
37 f1of 5413 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
4023ad2antrr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 𝐹:ω–onto𝐴)
4124ad2antrr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹𝑘) ≠ (𝐹𝑗))
42 simplr 520 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 𝑚 ∈ (ℤ‘0))
43 nn0uz 9467 . . . . . . . . . . . . . . . . 17 0 = (ℤ‘0)
4442, 43eleqtrrdi 2251 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 𝑚 ∈ ℕ0)
45 peano2nn0 9124 . . . . . . . . . . . . . . . 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 12120 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → dom (𝐻‘(𝑚 + 1)) ∈ ω)
4847adantr 274 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) ∈ ω)
4938, 48ffvelrnd 5602 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ∈ ℕ0)
5049nn0red 9138 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ∈ ℝ)
5144nn0red 9138 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 𝑚 ∈ ℝ)
5251adantr 274 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → 𝑚 ∈ ℝ)
53 peano2re 8005 . . . . . . . . . . . 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 12118 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝐻‘(𝑚 + 1)) = if((𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)), (𝐻𝑚), ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩})))
5655adantr 274 . . . . . . . . . . . . . . 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 3512 . . . . . . . . . . . . . . 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 2190 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝐻‘(𝑚 + 1)) = (𝐻𝑚))
6059dmeqd 4787 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) = dom (𝐻𝑚))
6160fveq2d 5471 . . . . . . . . . . . 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 9173 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 0 ∈ ℤ)
6439, 40, 41, 25, 26, 27, 28, 44ennnfonelemom 12120 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → dom (𝐻𝑚) ∈ ω)
65 f1ocnv 5426 . . . . . . . . . . . . . . . . . . . 20 (𝑁:ω–1-1-onto→ℕ0𝑁:ℕ01-1-onto→ω)
6636, 65ax-mp 5 . . . . . . . . . . . . . . . . . . 19 𝑁:ℕ01-1-onto→ω
67 f1of 5413 . . . . . . . . . . . . . . . . . . 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 5602 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ0 → (𝑁𝑚) ∈ ω)
7144, 70syl 14 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁𝑚) ∈ ω)
7263, 26, 64, 71frec2uzled 10321 . . . . . . . . . . . . . . 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 5725 . . . . . . . . . . . . . . 15 ((𝑁:ω–1-1-onto→ℕ0𝑚 ∈ ℕ0) → (𝑁‘(𝑁𝑚)) = 𝑚)
7536, 44, 74sylancr 411 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁‘(𝑁𝑚)) = 𝑚)
7673, 75breqtrd 3990 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁‘dom (𝐻𝑚)) ≤ 𝑚)
7776adantr 274 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻𝑚)) ≤ 𝑚)
7861, 77eqbrtrd 3986 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ 𝑚)
7952lep1d 8796 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → 𝑚 ≤ (𝑚 + 1))
8050, 52, 54, 78, 79letrd 7993 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑚 + 1))
81 f1ocnvfv2 5725 . . . . . . . . . . . 12 ((𝑁:ω–1-1-onto→ℕ0 ∧ (𝑚 + 1) ∈ ℕ0) → (𝑁‘(𝑁‘(𝑚 + 1))) = (𝑚 + 1))
8236, 46, 81sylancr 411 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁‘(𝑁‘(𝑚 + 1))) = (𝑚 + 1))
8382adantr 274 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘(𝑁‘(𝑚 + 1))) = (𝑚 + 1))
8480, 83breqtrrd 3992 . . . . . . . . 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 5602 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝑁‘(𝑚 + 1)) ∈ ω)
8763, 26, 47, 86frec2uzled 10321 . . . . . . . . . 10 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (dom (𝐻‘(𝑚 + 1)) ⊆ (𝑁‘(𝑚 + 1)) ↔ (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑁‘(𝑁‘(𝑚 + 1)))))
8887adantr 274 . . . . . . . . 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 274 . . . . . . . . . . . . . . . . . 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 3515 . . . . . . . . . . . . . . . . . 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 2190 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝐻‘(𝑚 + 1)) = ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩}))
9493dmeqd 4787 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) = dom ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩}))
95 dmun 4792 . . . . . . . . . . . . . . . 16 dom ((𝐻𝑚) ∪ {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩}) = (dom (𝐻𝑚) ∪ dom {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩})
9694, 95eqtrdi 2206 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) = (dom (𝐻𝑚) ∪ dom {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩}))
97 fof 5391 . . . . . . . . . . . . . . . . . . . 20 (𝐹:ω–onto𝐴𝐹:ω⟶𝐴)
9840, 97syl 14 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → 𝐹:ω⟶𝐴)
9998, 71ffvelrnd 5602 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → (𝐹‘(𝑁𝑚)) ∈ 𝐴)
10099adantr 274 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝐹‘(𝑁𝑚)) ∈ 𝐴)
101 dmsnopg 5056 . . . . . . . . . . . . . . . . 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 3261 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (dom (𝐻𝑚) ∪ dom {⟨dom (𝐻𝑚), (𝐹‘(𝑁𝑚))⟩}) = (dom (𝐻𝑚) ∪ {dom (𝐻𝑚)}))
10496, 103eqtrd 2190 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) = (dom (𝐻𝑚) ∪ {dom (𝐻𝑚)}))
105 df-suc 4331 . . . . . . . . . . . . . 14 suc dom (𝐻𝑚) = (dom (𝐻𝑚) ∪ {dom (𝐻𝑚)})
106104, 105eqtr4di 2208 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) = suc dom (𝐻𝑚))
107 simplr 520 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻𝑚) ⊆ (𝑁𝑚))
10871adantr 274 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁𝑚) ∈ ω)
109 nnsucsssuc 6436 . . . . . . . . . . . . . . 15 ((dom (𝐻𝑚) ∈ ω ∧ (𝑁𝑚) ∈ ω) → (dom (𝐻𝑚) ⊆ (𝑁𝑚) ↔ suc dom (𝐻𝑚) ⊆ suc (𝑁𝑚)))
11064, 108, 109syl2an2r 585 . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) ⊆ suc (𝑁𝑚))
113 0zd 9173 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → 0 ∈ ℤ)
11447adantr 274 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → dom (𝐻‘(𝑚 + 1)) ∈ ω)
115 peano2 4553 . . . . . . . . . . . . . 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 10321 . . . . . . . . . . . 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 10293 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘suc (𝑁𝑚)) = ((𝑁‘(𝑁𝑚)) + 1))
12075adantr 274 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘(𝑁𝑚)) = 𝑚)
121120oveq1d 5836 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → ((𝑁‘(𝑁𝑚)) + 1) = (𝑚 + 1))
122119, 121eqtrd 2190 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘suc (𝑁𝑚)) = (𝑚 + 1))
123118, 122breqtrd 3990 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑚 + 1))
12482adantr 274 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘(𝑁‘(𝑚 + 1))) = (𝑚 + 1))
125123, 124breqtrrd 3992 . . . . . . . . 9 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘dom (𝐻‘(𝑚 + 1))) ≤ (𝑁‘(𝑁‘(𝑚 + 1))))
12686adantr 274 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) ∧ ¬ (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚))) → (𝑁‘(𝑚 + 1)) ∈ ω)
127113, 26, 114, 126frec2uzled 10321 . . . . . . . . 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 12111 . . . . . . . . 9 (((𝜑𝑚 ∈ (ℤ‘0)) ∧ dom (𝐻𝑚) ⊆ (𝑁𝑚)) → DECID (𝐹‘(𝑁𝑚)) ∈ (𝐹 “ (𝑁𝑚)))
130 exmiddc 822 . . . . . . . . 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 788 . . . . . . 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 9493 . . 3 (𝑃 ∈ (ℤ‘0) → (𝜑 → dom (𝐻𝑃) ⊆ (𝑁𝑃)))
137136, 43eleq2s 2252 . 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 698  DECID wdc 820   = wceq 1335   ∈ wcel 2128   ≠ wne 2327  ∀wral 2435  ∃wrex 2436   ∪ cun 3100   ⊆ wss 3102  ∅c0 3394  ifcif 3505  {csn 3560  ⟨cop 3563   class class class wbr 3965   ↦ cmpt 4025  suc csuc 4325  ωcom 4548  ◡ccnv 4584  dom cdm 4585   “ cima 4588  ⟶wf 5165  –onto→wfo 5167  –1-1-onto→wf1o 5168  ‘cfv 5169  (class class class)co 5821   ∈ cmpo 5823  freccfrec 6334   ↑pm cpm 6591  ℝcr 7725  0cc0 7726  1c1 7727   + caddc 7729   ≤ cle 7907   − cmin 8040  ℕ0cn0 9084  ℤcz 9161  ℤ≥cuz 9433  seqcseq 10337 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546  ax-cnex 7817  ax-resscn 7818  ax-1cn 7819  ax-1re 7820  ax-icn 7821  ax-addcl 7822  ax-addrcl 7823  ax-mulcl 7824  ax-addcom 7826  ax-addass 7828  ax-distr 7830  ax-i2m1 7831  ax-0lt1 7832  ax-0id 7834  ax-rnegex 7835  ax-cnre 7837  ax-pre-ltirr 7838  ax-pre-ltwlin 7839  ax-pre-lttrn 7840  ax-pre-ltadd 7842 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-frec 6335  df-pm 6593  df-pnf 7908  df-mnf 7909  df-xr 7910  df-ltxr 7911  df-le 7912  df-sub 8042  df-neg 8043  df-inn 8828  df-n0 9085  df-z 9162  df-uz 9434  df-seqfrec 10338 This theorem is referenced by: (None)
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