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Theorem ctinfom 11783
Description: A condition for a set being countably infinite. Restates ennnfone 11780 in terms of ω and function image. Like ennnfone 11780 the condition can be summarized as 𝐴 being countable, infinite, and having decidable equality. (Contributed by Jim Kingdon, 7-Aug-2023.)
Assertion
Ref Expression
ctinfom (𝐴 ≈ ℕ ↔ (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))))
Distinct variable groups:   𝐴,𝑓,𝑛   𝑥,𝐴,𝑦   𝑓,𝑘,𝑛
Allowed substitution hint:   𝐴(𝑘)

Proof of Theorem ctinfom
Dummy variables 𝑎 𝑑 𝑖 𝑚 𝑔 𝑏 𝑐 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ennnfone 11780 . . . 4 (𝐴 ≈ ℕ ↔ (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑔(𝑔:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖))))
21simplbi 270 . . 3 (𝐴 ≈ ℕ → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
3 nnenom 10097 . . . . . . 7 ℕ ≈ ω
4 entr 6632 . . . . . . 7 ((𝐴 ≈ ℕ ∧ ℕ ≈ ω) → 𝐴 ≈ ω)
53, 4mpan2 419 . . . . . 6 (𝐴 ≈ ℕ → 𝐴 ≈ ω)
65ensymd 6631 . . . . 5 (𝐴 ≈ ℕ → ω ≈ 𝐴)
7 bren 6595 . . . . 5 (ω ≈ 𝐴 ↔ ∃𝑓 𝑓:ω–1-1-onto𝐴)
86, 7sylib 121 . . . 4 (𝐴 ≈ ℕ → ∃𝑓 𝑓:ω–1-1-onto𝐴)
9 f1ofo 5330 . . . . . . . 8 (𝑓:ω–1-1-onto𝐴𝑓:ω–onto𝐴)
109adantl 273 . . . . . . 7 ((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) → 𝑓:ω–onto𝐴)
11 simpr 109 . . . . . . . . 9 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ∈ ω)
12 nnord 4485 . . . . . . . . . . . 12 (𝑛 ∈ ω → Ord 𝑛)
1312adantl 273 . . . . . . . . . . 11 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → Ord 𝑛)
14 ordirr 4417 . . . . . . . . . . 11 (Ord 𝑛 → ¬ 𝑛𝑛)
1513, 14syl 14 . . . . . . . . . 10 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → ¬ 𝑛𝑛)
16 f1of1 5322 . . . . . . . . . . . 12 (𝑓:ω–1-1-onto𝐴𝑓:ω–1-1𝐴)
1716ad2antlr 478 . . . . . . . . . . 11 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → 𝑓:ω–1-1𝐴)
18 omelon 4482 . . . . . . . . . . . . 13 ω ∈ On
1918onelssi 4311 . . . . . . . . . . . 12 (𝑛 ∈ ω → 𝑛 ⊆ ω)
2019adantl 273 . . . . . . . . . . 11 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
21 f1elima 5628 . . . . . . . . . . 11 ((𝑓:ω–1-1𝐴𝑛 ∈ ω ∧ 𝑛 ⊆ ω) → ((𝑓𝑛) ∈ (𝑓𝑛) ↔ 𝑛𝑛))
2217, 11, 20, 21syl3anc 1199 . . . . . . . . . 10 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → ((𝑓𝑛) ∈ (𝑓𝑛) ↔ 𝑛𝑛))
2315, 22mtbird 645 . . . . . . . . 9 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → ¬ (𝑓𝑛) ∈ (𝑓𝑛))
24 fveq2 5375 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑓𝑘) = (𝑓𝑛))
2524eleq1d 2183 . . . . . . . . . . 11 (𝑘 = 𝑛 → ((𝑓𝑘) ∈ (𝑓𝑛) ↔ (𝑓𝑛) ∈ (𝑓𝑛)))
2625notbid 639 . . . . . . . . . 10 (𝑘 = 𝑛 → (¬ (𝑓𝑘) ∈ (𝑓𝑛) ↔ ¬ (𝑓𝑛) ∈ (𝑓𝑛)))
2726rspcev 2760 . . . . . . . . 9 ((𝑛 ∈ ω ∧ ¬ (𝑓𝑛) ∈ (𝑓𝑛)) → ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))
2811, 23, 27syl2anc 406 . . . . . . . 8 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))
2928ralrimiva 2479 . . . . . . 7 ((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))
3010, 29jca 302 . . . . . 6 ((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) → (𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)))
3130ex 114 . . . . 5 (𝐴 ≈ ℕ → (𝑓:ω–1-1-onto𝐴 → (𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))))
3231eximdv 1834 . . . 4 (𝐴 ≈ ℕ → (∃𝑓 𝑓:ω–1-1-onto𝐴 → ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))))
338, 32mpd 13 . . 3 (𝐴 ≈ ℕ → ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)))
342, 33jca 302 . 2 (𝐴 ≈ ℕ → (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))))
35 oveq1 5735 . . . . . . . . 9 (𝑏 = 𝑎 → (𝑏 + 1) = (𝑎 + 1))
3635cbvmptv 3984 . . . . . . . 8 (𝑏 ∈ ℤ ↦ (𝑏 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1))
37 freceq1 6243 . . . . . . . 8 ((𝑏 ∈ ℤ ↦ (𝑏 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1)) → frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) = frec((𝑎 ∈ ℤ ↦ (𝑎 + 1)), 0))
3836, 37ax-mp 7 . . . . . . 7 frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) = frec((𝑎 ∈ ℤ ↦ (𝑎 + 1)), 0)
39 eqid 2115 . . . . . . 7 (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))
40 simpl 108 . . . . . . 7 ((𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)) → 𝑓:ω–onto𝐴)
41 fveq2 5375 . . . . . . . . . . . . 13 (𝑘 = 𝑑 → (𝑓𝑘) = (𝑓𝑑))
4241eleq1d 2183 . . . . . . . . . . . 12 (𝑘 = 𝑑 → ((𝑓𝑘) ∈ (𝑓𝑛) ↔ (𝑓𝑑) ∈ (𝑓𝑛)))
4342notbid 639 . . . . . . . . . . 11 (𝑘 = 𝑑 → (¬ (𝑓𝑘) ∈ (𝑓𝑛) ↔ ¬ (𝑓𝑑) ∈ (𝑓𝑛)))
4443cbvrexv 2629 . . . . . . . . . 10 (∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑛))
4544ralbii 2415 . . . . . . . . 9 (∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛) ↔ ∀𝑛 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑛))
46 imaeq2 4835 . . . . . . . . . . . . 13 (𝑛 = 𝑐 → (𝑓𝑛) = (𝑓𝑐))
4746eleq2d 2184 . . . . . . . . . . . 12 (𝑛 = 𝑐 → ((𝑓𝑑) ∈ (𝑓𝑛) ↔ (𝑓𝑑) ∈ (𝑓𝑐)))
4847notbid 639 . . . . . . . . . . 11 (𝑛 = 𝑐 → (¬ (𝑓𝑑) ∈ (𝑓𝑛) ↔ ¬ (𝑓𝑑) ∈ (𝑓𝑐)))
4948rexbidv 2412 . . . . . . . . . 10 (𝑛 = 𝑐 → (∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑐)))
5049cbvralv 2628 . . . . . . . . 9 (∀𝑛 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑛) ↔ ∀𝑐 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑐))
5145, 50sylbb 122 . . . . . . . 8 (∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛) → ∀𝑐 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑐))
5251adantl 273 . . . . . . 7 ((𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)) → ∀𝑐 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑐))
5338, 39, 40, 52ctinfomlemom 11782 . . . . . 6 ((𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)) → ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)):ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)))
54 vex 2660 . . . . . . . 8 𝑓 ∈ V
55 frecex 6245 . . . . . . . . 9 frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
5655cnvex 5035 . . . . . . . 8 frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
5754, 56coex 5042 . . . . . . 7 (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) ∈ V
58 foeq1 5299 . . . . . . . 8 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔:ℕ0onto𝐴 ↔ (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)):ℕ0onto𝐴))
59 fveq1 5374 . . . . . . . . . . . 12 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔𝑗) = ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗))
60 fveq1 5374 . . . . . . . . . . . 12 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔𝑖) = ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖))
6159, 60neeq12d 2302 . . . . . . . . . . 11 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → ((𝑔𝑗) ≠ (𝑔𝑖) ↔ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)))
6261ralbidv 2411 . . . . . . . . . 10 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (∀𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖) ↔ ∀𝑖 ∈ (0...𝑚)((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)))
6362rexbidv 2412 . . . . . . . . 9 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (∃𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖) ↔ ∃𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)))
6463ralbidv 2411 . . . . . . . 8 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖) ↔ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)))
6558, 64anbi12d 462 . . . . . . 7 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → ((𝑔:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖)) ↔ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)):ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖))))
6657, 65spcev 2751 . . . . . 6 (((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)):ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)) → ∃𝑔(𝑔:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖)))
6753, 66syl 14 . . . . 5 ((𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)) → ∃𝑔(𝑔:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖)))
6867exlimiv 1560 . . . 4 (∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)) → ∃𝑔(𝑔:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖)))
6968anim2i 337 . . 3 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))) → (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑔(𝑔:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖))))
7069, 1sylibr 133 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))) → 𝐴 ≈ ℕ)
7134, 70impbii 125 1 (𝐴 ≈ ℕ ↔ (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wb 104  DECID wdc 802   = wceq 1314  wex 1451  wcel 1463  wne 2282  wral 2390  wrex 2391  wss 3037   class class class wbr 3895  cmpt 3949  Ord word 4244  ωcom 4464  ccnv 4498  cima 4502  ccom 4503  1-1wf1 5078  ontowfo 5079  1-1-ontowf1o 5080  cfv 5081  (class class class)co 5728  freccfrec 6241  cen 6586  0cc0 7544  1c1 7545   + caddc 7547  cn 8627  0cn0 8878  cz 8955  ...cfz 9680
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 7633  ax-resscn 7634  ax-1cn 7635  ax-1re 7636  ax-icn 7637  ax-addcl 7638  ax-addrcl 7639  ax-mulcl 7640  ax-addcom 7642  ax-addass 7644  ax-distr 7646  ax-i2m1 7647  ax-0lt1 7648  ax-0id 7650  ax-rnegex 7651  ax-cnre 7653  ax-pre-ltirr 7654  ax-pre-ltwlin 7655  ax-pre-lttrn 7656  ax-pre-ltadd 7658
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-er 6383  df-pm 6499  df-en 6589  df-pnf 7723  df-mnf 7724  df-xr 7725  df-ltxr 7726  df-le 7727  df-sub 7855  df-neg 7856  df-inn 8628  df-n0 8879  df-z 8956  df-uz 9226  df-fz 9681  df-seqfrec 10109
This theorem is referenced by:  ctinf  11785
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