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Theorem curfpropd 17479
 Description: If two categories have the same set of objects, morphisms, and compositions, then they curry the same functor to the same result. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
curfpropd.1 (𝜑 → (Homf𝐴) = (Homf𝐵))
curfpropd.2 (𝜑 → (compf𝐴) = (compf𝐵))
curfpropd.3 (𝜑 → (Homf𝐶) = (Homf𝐷))
curfpropd.4 (𝜑 → (compf𝐶) = (compf𝐷))
curfpropd.a (𝜑𝐴 ∈ Cat)
curfpropd.b (𝜑𝐵 ∈ Cat)
curfpropd.c (𝜑𝐶 ∈ Cat)
curfpropd.d (𝜑𝐷 ∈ Cat)
curfpropd.f (𝜑𝐹 ∈ ((𝐴 ×c 𝐶) Func 𝐸))
Assertion
Ref Expression
curfpropd (𝜑 → (⟨𝐴, 𝐶⟩ curryF 𝐹) = (⟨𝐵, 𝐷⟩ curryF 𝐹))

Proof of Theorem curfpropd
Dummy variables 𝑥 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 curfpropd.1 . . . . 5 (𝜑 → (Homf𝐴) = (Homf𝐵))
21homfeqbas 16962 . . . 4 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
3 curfpropd.3 . . . . . . . 8 (𝜑 → (Homf𝐶) = (Homf𝐷))
43homfeqbas 16962 . . . . . . 7 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
54adantr 484 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐴)) → (Base‘𝐶) = (Base‘𝐷))
65mpteq1d 5141 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐴)) → (𝑦 ∈ (Base‘𝐶) ↦ (𝑥(1st𝐹)𝑦)) = (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)))
75adantr 484 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
8 eqid 2824 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
9 eqid 2824 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
10 eqid 2824 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
113ad2antrr 725 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (Homf𝐶) = (Homf𝐷))
12 simprl 770 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
13 simprr 772 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶))
148, 9, 10, 11, 12, 13homfeqval 16963 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦(Hom ‘𝐷)𝑧))
15 curfpropd.2 . . . . . . . . . . 11 (𝜑 → (compf𝐴) = (compf𝐵))
16 curfpropd.a . . . . . . . . . . 11 (𝜑𝐴 ∈ Cat)
17 curfpropd.b . . . . . . . . . . 11 (𝜑𝐵 ∈ Cat)
181, 15, 16, 17cidpropd 16976 . . . . . . . . . 10 (𝜑 → (Id‘𝐴) = (Id‘𝐵))
1918ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (Id‘𝐴) = (Id‘𝐵))
2019fveq1d 6660 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ((Id‘𝐴)‘𝑥) = ((Id‘𝐵)‘𝑥))
2120oveq1d 7160 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (((Id‘𝐴)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔) = (((Id‘𝐵)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔))
2214, 21mpteq12dv 5137 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((Id‘𝐴)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐵)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))
235, 7, 22mpoeq123dva 7217 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐴)) → (𝑦 ∈ (Base‘𝐶), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((Id‘𝐴)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔))) = (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐵)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔))))
246, 23opeq12d 4797 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐴)) → ⟨(𝑦 ∈ (Base‘𝐶) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐶), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((Id‘𝐴)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩ = ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐵)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩)
252, 24mpteq12dva 5136 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐴) ↦ ⟨(𝑦 ∈ (Base‘𝐶) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐶), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((Id‘𝐴)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩) = (𝑥 ∈ (Base‘𝐵) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐵)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))
262adantr 484 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐴)) → (Base‘𝐴) = (Base‘𝐵))
27 eqid 2824 . . . . . 6 (Base‘𝐴) = (Base‘𝐴)
28 eqid 2824 . . . . . 6 (Hom ‘𝐴) = (Hom ‘𝐴)
29 eqid 2824 . . . . . 6 (Hom ‘𝐵) = (Hom ‘𝐵)
301adantr 484 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (Homf𝐴) = (Homf𝐵))
31 simprl 770 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝐴))
32 simprr 772 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐴))
3327, 28, 29, 30, 31, 32homfeqval 16963 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
344ad2antrr 725 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦)) → (Base‘𝐶) = (Base‘𝐷))
35 curfpropd.4 . . . . . . . . . 10 (𝜑 → (compf𝐶) = (compf𝐷))
36 curfpropd.c . . . . . . . . . 10 (𝜑𝐶 ∈ Cat)
37 curfpropd.d . . . . . . . . . 10 (𝜑𝐷 ∈ Cat)
383, 35, 36, 37cidpropd 16976 . . . . . . . . 9 (𝜑 → (Id‘𝐶) = (Id‘𝐷))
3938ad3antrrr 729 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦)) ∧ 𝑧 ∈ (Base‘𝐶)) → (Id‘𝐶) = (Id‘𝐷))
4039fveq1d 6660 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦)) ∧ 𝑧 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑧) = ((Id‘𝐷)‘𝑧))
4140oveq2d 7161 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦)) ∧ 𝑧 ∈ (Base‘𝐶)) → (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐶)‘𝑧)) = (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))
4234, 41mpteq12dva 5136 . . . . 5 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦)) → (𝑧 ∈ (Base‘𝐶) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐶)‘𝑧))) = (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))
4333, 42mpteq12dva 5136 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦) ↦ (𝑧 ∈ (Base‘𝐶) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐶)‘𝑧)))) = (𝑔 ∈ (𝑥(Hom ‘𝐵)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))
442, 26, 43mpoeq123dva 7217 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐴), 𝑦 ∈ (Base‘𝐴) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦) ↦ (𝑧 ∈ (Base‘𝐶) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐶)‘𝑧))))) = (𝑥 ∈ (Base‘𝐵), 𝑦 ∈ (Base‘𝐵) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐵)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))))
4525, 44opeq12d 4797 . 2 (𝜑 → ⟨(𝑥 ∈ (Base‘𝐴) ↦ ⟨(𝑦 ∈ (Base‘𝐶) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐶), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((Id‘𝐴)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐴), 𝑦 ∈ (Base‘𝐴) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦) ↦ (𝑧 ∈ (Base‘𝐶) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐶)‘𝑧)))))⟩ = ⟨(𝑥 ∈ (Base‘𝐵) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐵)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐵), 𝑦 ∈ (Base‘𝐵) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐵)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩)
46 eqid 2824 . . 3 (⟨𝐴, 𝐶⟩ curryF 𝐹) = (⟨𝐴, 𝐶⟩ curryF 𝐹)
47 curfpropd.f . . 3 (𝜑𝐹 ∈ ((𝐴 ×c 𝐶) Func 𝐸))
48 eqid 2824 . . 3 (Id‘𝐴) = (Id‘𝐴)
49 eqid 2824 . . 3 (Id‘𝐶) = (Id‘𝐶)
5046, 27, 16, 36, 47, 8, 9, 48, 28, 49curfval 17469 . 2 (𝜑 → (⟨𝐴, 𝐶⟩ curryF 𝐹) = ⟨(𝑥 ∈ (Base‘𝐴) ↦ ⟨(𝑦 ∈ (Base‘𝐶) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐶), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((Id‘𝐴)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐴), 𝑦 ∈ (Base‘𝐴) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦) ↦ (𝑧 ∈ (Base‘𝐶) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐶)‘𝑧)))))⟩)
51 eqid 2824 . . 3 (⟨𝐵, 𝐷⟩ curryF 𝐹) = (⟨𝐵, 𝐷⟩ curryF 𝐹)
52 eqid 2824 . . 3 (Base‘𝐵) = (Base‘𝐵)
531, 15, 3, 35, 16, 17, 36, 37xpcpropd 17454 . . . . 5 (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))
5453oveq1d 7160 . . . 4 (𝜑 → ((𝐴 ×c 𝐶) Func 𝐸) = ((𝐵 ×c 𝐷) Func 𝐸))
5547, 54eleqtrd 2918 . . 3 (𝜑𝐹 ∈ ((𝐵 ×c 𝐷) Func 𝐸))
56 eqid 2824 . . 3 (Base‘𝐷) = (Base‘𝐷)
57 eqid 2824 . . 3 (Id‘𝐵) = (Id‘𝐵)
58 eqid 2824 . . 3 (Id‘𝐷) = (Id‘𝐷)
5951, 52, 17, 37, 55, 56, 10, 57, 29, 58curfval 17469 . 2 (𝜑 → (⟨𝐵, 𝐷⟩ curryF 𝐹) = ⟨(𝑥 ∈ (Base‘𝐵) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐵)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐵), 𝑦 ∈ (Base‘𝐵) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐵)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩)
6045, 50, 593eqtr4d 2869 1 (𝜑 → (⟨𝐴, 𝐶⟩ curryF 𝐹) = (⟨𝐵, 𝐷⟩ curryF 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ⟨cop 4555   ↦ cmpt 5132  ‘cfv 6343  (class class class)co 7145   ∈ cmpo 7147  1st c1st 7677  2nd c2nd 7678  Basecbs 16479  Hom chom 16572  Catccat 16931  Idccid 16932  Homf chomf 16933  compfccomf 16934   Func cfunc 17120   ×c cxpc 17414   curryF ccurf 17456 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-tp 4554  df-op 4556  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7571  df-1st 7679  df-2nd 7680  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-1o 8092  df-oadd 8096  df-er 8279  df-en 8500  df-dom 8501  df-sdom 8502  df-fin 8503  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11693  df-3 11694  df-4 11695  df-5 11696  df-6 11697  df-7 11698  df-8 11699  df-9 11700  df-n0 11891  df-z 11975  df-dec 12092  df-uz 12237  df-fz 12891  df-struct 16481  df-ndx 16482  df-slot 16483  df-base 16485  df-hom 16585  df-cco 16586  df-cat 16935  df-cid 16936  df-homf 16937  df-comf 16938  df-xpc 17418  df-curf 17460 This theorem is referenced by:  yonpropd  17514  oppcyon  17515
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