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Theorem ctinfom 11975
Description: A condition for a set being countably infinite. Restates ennnfone 11972 in terms of ω and function image. Like ennnfone 11972 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 11972 . . . 4 (𝐴 ≈ ℕ ↔ (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑔(𝑔:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖))))
21simplbi 272 . . 3 (𝐴 ≈ ℕ → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
3 nnenom 10237 . . . . . . 7 ℕ ≈ ω
4 entr 6685 . . . . . . 7 ((𝐴 ≈ ℕ ∧ ℕ ≈ ω) → 𝐴 ≈ ω)
53, 4mpan2 422 . . . . . 6 (𝐴 ≈ ℕ → 𝐴 ≈ ω)
65ensymd 6684 . . . . 5 (𝐴 ≈ ℕ → ω ≈ 𝐴)
7 bren 6648 . . . . 5 (ω ≈ 𝐴 ↔ ∃𝑓 𝑓:ω–1-1-onto𝐴)
86, 7sylib 121 . . . 4 (𝐴 ≈ ℕ → ∃𝑓 𝑓:ω–1-1-onto𝐴)
9 f1ofo 5381 . . . . . . . 8 (𝑓:ω–1-1-onto𝐴𝑓:ω–onto𝐴)
109adantl 275 . . . . . . 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 4532 . . . . . . . . . . . 12 (𝑛 ∈ ω → Ord 𝑛)
1312adantl 275 . . . . . . . . . . 11 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → Ord 𝑛)
14 ordirr 4464 . . . . . . . . . . 11 (Ord 𝑛 → ¬ 𝑛𝑛)
1513, 14syl 14 . . . . . . . . . 10 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → ¬ 𝑛𝑛)
16 f1of1 5373 . . . . . . . . . . . 12 (𝑓:ω–1-1-onto𝐴𝑓:ω–1-1𝐴)
1716ad2antlr 481 . . . . . . . . . . 11 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → 𝑓:ω–1-1𝐴)
18 omelon 4529 . . . . . . . . . . . . 13 ω ∈ On
1918onelssi 4358 . . . . . . . . . . . 12 (𝑛 ∈ ω → 𝑛 ⊆ ω)
2019adantl 275 . . . . . . . . . . 11 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
21 f1elima 5681 . . . . . . . . . . 11 ((𝑓:ω–1-1𝐴𝑛 ∈ ω ∧ 𝑛 ⊆ ω) → ((𝑓𝑛) ∈ (𝑓𝑛) ↔ 𝑛𝑛))
2217, 11, 20, 21syl3anc 1217 . . . . . . . . . 10 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → ((𝑓𝑛) ∈ (𝑓𝑛) ↔ 𝑛𝑛))
2315, 22mtbird 663 . . . . . . . . 9 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → ¬ (𝑓𝑛) ∈ (𝑓𝑛))
24 fveq2 5428 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑓𝑘) = (𝑓𝑛))
2524eleq1d 2209 . . . . . . . . . . 11 (𝑘 = 𝑛 → ((𝑓𝑘) ∈ (𝑓𝑛) ↔ (𝑓𝑛) ∈ (𝑓𝑛)))
2625notbid 657 . . . . . . . . . 10 (𝑘 = 𝑛 → (¬ (𝑓𝑘) ∈ (𝑓𝑛) ↔ ¬ (𝑓𝑛) ∈ (𝑓𝑛)))
2726rspcev 2792 . . . . . . . . 9 ((𝑛 ∈ ω ∧ ¬ (𝑓𝑛) ∈ (𝑓𝑛)) → ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))
2811, 23, 27syl2anc 409 . . . . . . . 8 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))
2928ralrimiva 2508 . . . . . . 7 ((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))
3010, 29jca 304 . . . . . 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 1853 . . . 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 304 . 2 (𝐴 ≈ ℕ → (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))))
35 oveq1 5788 . . . . . . . . 9 (𝑏 = 𝑎 → (𝑏 + 1) = (𝑎 + 1))
3635cbvmptv 4031 . . . . . . . 8 (𝑏 ∈ ℤ ↦ (𝑏 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1))
37 freceq1 6296 . . . . . . . 8 ((𝑏 ∈ ℤ ↦ (𝑏 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1)) → frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) = frec((𝑎 ∈ ℤ ↦ (𝑎 + 1)), 0))
3836, 37ax-mp 5 . . . . . . 7 frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) = frec((𝑎 ∈ ℤ ↦ (𝑎 + 1)), 0)
39 eqid 2140 . . . . . . 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 5428 . . . . . . . . . . . . 13 (𝑘 = 𝑑 → (𝑓𝑘) = (𝑓𝑑))
4241eleq1d 2209 . . . . . . . . . . . 12 (𝑘 = 𝑑 → ((𝑓𝑘) ∈ (𝑓𝑛) ↔ (𝑓𝑑) ∈ (𝑓𝑛)))
4342notbid 657 . . . . . . . . . . 11 (𝑘 = 𝑑 → (¬ (𝑓𝑘) ∈ (𝑓𝑛) ↔ ¬ (𝑓𝑑) ∈ (𝑓𝑛)))
4443cbvrexv 2658 . . . . . . . . . 10 (∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑛))
4544ralbii 2444 . . . . . . . . 9 (∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛) ↔ ∀𝑛 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑛))
46 imaeq2 4884 . . . . . . . . . . . . 13 (𝑛 = 𝑐 → (𝑓𝑛) = (𝑓𝑐))
4746eleq2d 2210 . . . . . . . . . . . 12 (𝑛 = 𝑐 → ((𝑓𝑑) ∈ (𝑓𝑛) ↔ (𝑓𝑑) ∈ (𝑓𝑐)))
4847notbid 657 . . . . . . . . . . 11 (𝑛 = 𝑐 → (¬ (𝑓𝑑) ∈ (𝑓𝑛) ↔ ¬ (𝑓𝑑) ∈ (𝑓𝑐)))
4948rexbidv 2439 . . . . . . . . . 10 (𝑛 = 𝑐 → (∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑐)))
5049cbvralv 2657 . . . . . . . . 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 275 . . . . . . 7 ((𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)) → ∀𝑐 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑐))
5338, 39, 40, 52ctinfomlemom 11974 . . . . . 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 2692 . . . . . . . 8 𝑓 ∈ V
55 frecex 6298 . . . . . . . . 9 frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
5655cnvex 5084 . . . . . . . 8 frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
5754, 56coex 5091 . . . . . . 7 (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) ∈ V
58 foeq1 5348 . . . . . . . 8 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔:ℕ0onto𝐴 ↔ (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)):ℕ0onto𝐴))
59 fveq1 5427 . . . . . . . . . . . 12 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔𝑗) = ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗))
60 fveq1 5427 . . . . . . . . . . . 12 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔𝑖) = ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖))
6159, 60neeq12d 2329 . . . . . . . . . . 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 2438 . . . . . . . . . 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 2439 . . . . . . . . 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 2438 . . . . . . . 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 465 . . . . . . 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 2783 . . . . . 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 1578 . . . 4 (∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)) → ∃𝑔(𝑔:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖)))
6968anim2i 340 . . 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 820   = wceq 1332  wex 1469  wcel 1481  wne 2309  wral 2417  wrex 2418  wss 3075   class class class wbr 3936  cmpt 3996  Ord word 4291  ωcom 4511  ccnv 4545  cima 4549  ccom 4550  1-1wf1 5127  ontowfo 5128  1-1-ontowf1o 5129  cfv 5130  (class class class)co 5781  freccfrec 6294  cen 6639  0cc0 7643  1c1 7644   + caddc 7646  cn 8743  0cn0 9000  cz 9077  ...cfz 9820
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-addcom 7743  ax-addass 7745  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-0id 7751  ax-rnegex 7752  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-ltadd 7759
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-frec 6295  df-er 6436  df-pm 6552  df-en 6642  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-inn 8744  df-n0 9001  df-z 9078  df-uz 9350  df-fz 9821  df-seqfrec 10249
This theorem is referenced by:  ctinf  11977
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