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Theorem frecuzrdgtcl 10192
Description: The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10179 for the description of 𝐺 as the mapping from ω to (ℤ𝐶). (Contributed by Jim Kingdon, 26-May-2020.)
Hypotheses
Ref Expression
frec2uz.1 (𝜑𝐶 ∈ ℤ)
frec2uz.2 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
frecuzrdgrrn.a (𝜑𝐴𝑆)
frecuzrdgrrn.f ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
frecuzrdgrrn.2 𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
frecuzrdgtcl.3 (𝜑𝑇 = ran 𝑅)
Assertion
Ref Expression
frecuzrdgtcl (𝜑𝑇:(ℤ𝐶)⟶𝑆)
Distinct variable groups:   𝑦,𝐴   𝑥,𝐶,𝑦   𝑦,𝐺   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝑅(𝑥,𝑦)   𝑇(𝑥,𝑦)   𝐺(𝑥)

Proof of Theorem frecuzrdgtcl
Dummy variables 𝑤 𝑧 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frecuzrdgtcl.3 . . . . . . . . . 10 (𝜑𝑇 = ran 𝑅)
21eleq2d 2209 . . . . . . . . 9 (𝜑 → (𝑧𝑇𝑧 ∈ ran 𝑅))
3 frec2uz.1 . . . . . . . . . . 11 (𝜑𝐶 ∈ ℤ)
4 frec2uz.2 . . . . . . . . . . 11 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
5 frecuzrdgrrn.a . . . . . . . . . . 11 (𝜑𝐴𝑆)
6 frecuzrdgrrn.f . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
7 frecuzrdgrrn.2 . . . . . . . . . . 11 𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
83, 4, 5, 6, 7frecuzrdgrcl 10190 . . . . . . . . . 10 (𝜑𝑅:ω⟶((ℤ𝐶) × 𝑆))
9 ffn 5272 . . . . . . . . . 10 (𝑅:ω⟶((ℤ𝐶) × 𝑆) → 𝑅 Fn ω)
10 fvelrnb 5469 . . . . . . . . . 10 (𝑅 Fn ω → (𝑧 ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = 𝑧))
118, 9, 103syl 17 . . . . . . . . 9 (𝜑 → (𝑧 ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = 𝑧))
122, 11bitrd 187 . . . . . . . 8 (𝜑 → (𝑧𝑇 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = 𝑧))
133, 4, 5, 6, 7frecuzrdgrrn 10188 . . . . . . . . . 10 ((𝜑𝑤 ∈ ω) → (𝑅𝑤) ∈ ((ℤ𝐶) × 𝑆))
14 eleq1 2202 . . . . . . . . . 10 ((𝑅𝑤) = 𝑧 → ((𝑅𝑤) ∈ ((ℤ𝐶) × 𝑆) ↔ 𝑧 ∈ ((ℤ𝐶) × 𝑆)))
1513, 14syl5ibcom 154 . . . . . . . . 9 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = 𝑧𝑧 ∈ ((ℤ𝐶) × 𝑆)))
1615rexlimdva 2549 . . . . . . . 8 (𝜑 → (∃𝑤 ∈ ω (𝑅𝑤) = 𝑧𝑧 ∈ ((ℤ𝐶) × 𝑆)))
1712, 16sylbid 149 . . . . . . 7 (𝜑 → (𝑧𝑇𝑧 ∈ ((ℤ𝐶) × 𝑆)))
1817ssrdv 3103 . . . . . 6 (𝜑𝑇 ⊆ ((ℤ𝐶) × 𝑆))
19 xpss 4647 . . . . . 6 ((ℤ𝐶) × 𝑆) ⊆ (V × V)
2018, 19sstrdi 3109 . . . . 5 (𝜑𝑇 ⊆ (V × V))
21 df-rel 4546 . . . . 5 (Rel 𝑇𝑇 ⊆ (V × V))
2220, 21sylibr 133 . . . 4 (𝜑 → Rel 𝑇)
233, 4frec2uzf1od 10186 . . . . . . . . . . 11 (𝜑𝐺:ω–1-1-onto→(ℤ𝐶))
24 f1ocnvdm 5682 . . . . . . . . . . 11 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝑣 ∈ (ℤ𝐶)) → (𝐺𝑣) ∈ ω)
2523, 24sylan 281 . . . . . . . . . 10 ((𝜑𝑣 ∈ (ℤ𝐶)) → (𝐺𝑣) ∈ ω)
263, 4, 5, 6, 7frecuzrdgrrn 10188 . . . . . . . . . 10 ((𝜑 ∧ (𝐺𝑣) ∈ ω) → (𝑅‘(𝐺𝑣)) ∈ ((ℤ𝐶) × 𝑆))
2725, 26syldan 280 . . . . . . . . 9 ((𝜑𝑣 ∈ (ℤ𝐶)) → (𝑅‘(𝐺𝑣)) ∈ ((ℤ𝐶) × 𝑆))
28 xp2nd 6064 . . . . . . . . 9 ((𝑅‘(𝐺𝑣)) ∈ ((ℤ𝐶) × 𝑆) → (2nd ‘(𝑅‘(𝐺𝑣))) ∈ 𝑆)
2927, 28syl 14 . . . . . . . 8 ((𝜑𝑣 ∈ (ℤ𝐶)) → (2nd ‘(𝑅‘(𝐺𝑣))) ∈ 𝑆)
301eleq2d 2209 . . . . . . . . . . . 12 (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑇 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅))
31 fvelrnb 5469 . . . . . . . . . . . . 13 (𝑅 Fn ω → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩))
328, 9, 313syl 17 . . . . . . . . . . . 12 (𝜑 → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩))
3330, 32bitrd 187 . . . . . . . . . . 11 (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑇 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩))
343adantr 274 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ω) → 𝐶 ∈ ℤ)
355adantr 274 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ω) → 𝐴𝑆)
366adantlr 468 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤 ∈ ω) ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
37 simpr 109 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ω) → 𝑤 ∈ ω)
3834, 4, 35, 36, 7, 37frec2uzrdg 10189 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ ω) → (𝑅𝑤) = ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩)
3938eqeq1d 2148 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩))
40 vex 2689 . . . . . . . . . . . . . . . . . . . 20 𝑣 ∈ V
41 vex 2689 . . . . . . . . . . . . . . . . . . . 20 𝑧 ∈ V
4240, 41opth2 4162 . . . . . . . . . . . . . . . . . . 19 (⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩ ↔ ((𝐺𝑤) = 𝑣 ∧ (2nd ‘(𝑅𝑤)) = 𝑧))
4342simplbi 272 . . . . . . . . . . . . . . . . . 18 (⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣)
4439, 43syl6bi 162 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣))
45 f1ocnvfv 5680 . . . . . . . . . . . . . . . . . 18 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝑤 ∈ ω) → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4623, 45sylan 281 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ ω) → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4744, 46syld 45 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺𝑣) = 𝑤))
48 fveq2 5421 . . . . . . . . . . . . . . . . 17 ((𝐺𝑣) = 𝑤 → (𝑅‘(𝐺𝑣)) = (𝑅𝑤))
4948fveq2d 5425 . . . . . . . . . . . . . . . 16 ((𝐺𝑣) = 𝑤 → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
5047, 49syl6 33 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤))))
5150imp 123 . . . . . . . . . . . . . 14 (((𝜑𝑤 ∈ ω) ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
5240, 41op2ndd 6047 . . . . . . . . . . . . . . 15 ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅𝑤)) = 𝑧)
5352adantl 275 . . . . . . . . . . . . . 14 (((𝜑𝑤 ∈ ω) ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅𝑤)) = 𝑧)
5451, 53eqtr2d 2173 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ω) ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5554ex 114 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5655rexlimdva 2549 . . . . . . . . . . 11 (𝜑 → (∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5733, 56sylbid 149 . . . . . . . . . 10 (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑇𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5857alrimiv 1846 . . . . . . . . 9 (𝜑 → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5958adantr 274 . . . . . . . 8 ((𝜑𝑣 ∈ (ℤ𝐶)) → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
60 eqeq2 2149 . . . . . . . . . . 11 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (𝑧 = 𝑤𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
6160imbi2d 229 . . . . . . . . . 10 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ 𝑇𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ 𝑇𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
6261albidv 1796 . . . . . . . . 9 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇𝑧 = 𝑤) ↔ ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
6362spcegv 2774 . . . . . . . 8 ((2nd ‘(𝑅‘(𝐺𝑣))) ∈ 𝑆 → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))) → ∃𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇𝑧 = 𝑤)))
6429, 59, 63sylc 62 . . . . . . 7 ((𝜑𝑣 ∈ (ℤ𝐶)) → ∃𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇𝑧 = 𝑤))
65 nfv 1508 . . . . . . . 8 𝑤𝑣, 𝑧⟩ ∈ 𝑇
6665mo2r 2051 . . . . . . 7 (∃𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇𝑧 = 𝑤) → ∃*𝑧𝑣, 𝑧⟩ ∈ 𝑇)
6764, 66syl 14 . . . . . 6 ((𝜑𝑣 ∈ (ℤ𝐶)) → ∃*𝑧𝑣, 𝑧⟩ ∈ 𝑇)
68 dmss 4738 . . . . . . . . . . 11 (𝑇 ⊆ ((ℤ𝐶) × 𝑆) → dom 𝑇 ⊆ dom ((ℤ𝐶) × 𝑆))
6918, 68syl 14 . . . . . . . . . 10 (𝜑 → dom 𝑇 ⊆ dom ((ℤ𝐶) × 𝑆))
70 dmxpss 4969 . . . . . . . . . 10 dom ((ℤ𝐶) × 𝑆) ⊆ (ℤ𝐶)
7169, 70sstrdi 3109 . . . . . . . . 9 (𝜑 → dom 𝑇 ⊆ (ℤ𝐶))
723adantr 274 . . . . . . . . . . . . . 14 ((𝜑𝑣 ∈ (ℤ𝐶)) → 𝐶 ∈ ℤ)
735adantr 274 . . . . . . . . . . . . . 14 ((𝜑𝑣 ∈ (ℤ𝐶)) → 𝐴𝑆)
746adantlr 468 . . . . . . . . . . . . . 14 (((𝜑𝑣 ∈ (ℤ𝐶)) ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
75 simpr 109 . . . . . . . . . . . . . 14 ((𝜑𝑣 ∈ (ℤ𝐶)) → 𝑣 ∈ (ℤ𝐶))
7672, 4, 73, 74, 7, 75frecuzrdglem 10191 . . . . . . . . . . . . 13 ((𝜑𝑣 ∈ (ℤ𝐶)) → ⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ ran 𝑅)
771eleq2d 2209 . . . . . . . . . . . . . 14 (𝜑 → (⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ 𝑇 ↔ ⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ ran 𝑅))
7877adantr 274 . . . . . . . . . . . . 13 ((𝜑𝑣 ∈ (ℤ𝐶)) → (⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ 𝑇 ↔ ⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ ran 𝑅))
7976, 78mpbird 166 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ (ℤ𝐶)) → ⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ 𝑇)
80 opeldmg 4744 . . . . . . . . . . . . 13 ((𝑣 ∈ V ∧ (2nd ‘(𝑅‘(𝐺𝑣))) ∈ 𝑆) → (⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ 𝑇𝑣 ∈ dom 𝑇))
8140, 80mpan 420 . . . . . . . . . . . 12 ((2nd ‘(𝑅‘(𝐺𝑣))) ∈ 𝑆 → (⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ 𝑇𝑣 ∈ dom 𝑇))
8229, 79, 81sylc 62 . . . . . . . . . . 11 ((𝜑𝑣 ∈ (ℤ𝐶)) → 𝑣 ∈ dom 𝑇)
8382ex 114 . . . . . . . . . 10 (𝜑 → (𝑣 ∈ (ℤ𝐶) → 𝑣 ∈ dom 𝑇))
8483ssrdv 3103 . . . . . . . . 9 (𝜑 → (ℤ𝐶) ⊆ dom 𝑇)
8571, 84eqssd 3114 . . . . . . . 8 (𝜑 → dom 𝑇 = (ℤ𝐶))
8685eleq2d 2209 . . . . . . 7 (𝜑 → (𝑣 ∈ dom 𝑇𝑣 ∈ (ℤ𝐶)))
8786pm5.32i 449 . . . . . 6 ((𝜑𝑣 ∈ dom 𝑇) ↔ (𝜑𝑣 ∈ (ℤ𝐶)))
88 df-br 3930 . . . . . . 7 (𝑣𝑇𝑧 ↔ ⟨𝑣, 𝑧⟩ ∈ 𝑇)
8988mobii 2036 . . . . . 6 (∃*𝑧 𝑣𝑇𝑧 ↔ ∃*𝑧𝑣, 𝑧⟩ ∈ 𝑇)
9067, 87, 893imtr4i 200 . . . . 5 ((𝜑𝑣 ∈ dom 𝑇) → ∃*𝑧 𝑣𝑇𝑧)
9190ralrimiva 2505 . . . 4 (𝜑 → ∀𝑣 ∈ dom 𝑇∃*𝑧 𝑣𝑇𝑧)
92 dffun7 5150 . . . 4 (Fun 𝑇 ↔ (Rel 𝑇 ∧ ∀𝑣 ∈ dom 𝑇∃*𝑧 𝑣𝑇𝑧))
9322, 91, 92sylanbrc 413 . . 3 (𝜑 → Fun 𝑇)
94 df-fn 5126 . . 3 (𝑇 Fn (ℤ𝐶) ↔ (Fun 𝑇 ∧ dom 𝑇 = (ℤ𝐶)))
9593, 85, 94sylanbrc 413 . 2 (𝜑𝑇 Fn (ℤ𝐶))
96 rnss 4769 . . . 4 (𝑇 ⊆ ((ℤ𝐶) × 𝑆) → ran 𝑇 ⊆ ran ((ℤ𝐶) × 𝑆))
9718, 96syl 14 . . 3 (𝜑 → ran 𝑇 ⊆ ran ((ℤ𝐶) × 𝑆))
98 rnxpss 4970 . . 3 ran ((ℤ𝐶) × 𝑆) ⊆ 𝑆
9997, 98sstrdi 3109 . 2 (𝜑 → ran 𝑇𝑆)
100 df-f 5127 . 2 (𝑇:(ℤ𝐶)⟶𝑆 ↔ (𝑇 Fn (ℤ𝐶) ∧ ran 𝑇𝑆))
10195, 99, 100sylanbrc 413 1 (𝜑𝑇:(ℤ𝐶)⟶𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1329   = wceq 1331  wex 1468  wcel 1480  ∃*wmo 2000  wral 2416  wrex 2417  Vcvv 2686  wss 3071  cop 3530   class class class wbr 3929  cmpt 3989  ωcom 4504   × cxp 4537  ccnv 4538  dom cdm 4539  ran crn 4540  Rel wrel 4544  Fun wfun 5117   Fn wfn 5118  wf 5119  1-1-ontowf1o 5122  cfv 5123  (class class class)co 5774  cmpo 5776  2nd c2nd 6037  freccfrec 6287  1c1 7628   + caddc 7630  cz 9061  cuz 9333
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-ltadd 7743
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-inn 8728  df-n0 8985  df-z 9062  df-uz 9334
This theorem is referenced by:  frecuzrdg0  10193  frecuzrdgsuc  10194
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