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Theorem curfpropd 17979
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 17433 . . . 4 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
3 curfpropd.3 . . . . . . . 8 (𝜑 → (Homf𝐶) = (Homf𝐷))
43homfeqbas 17433 . . . . . . 7 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
54adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐴)) → (Base‘𝐶) = (Base‘𝐷))
65mpteq1d 5172 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐴)) → (𝑦 ∈ (Base‘𝐶) ↦ (𝑥(1st𝐹)𝑦)) = (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)))
75adantr 480 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
8 eqid 2733 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
9 eqid 2733 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
10 eqid 2733 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
113ad2antrr 722 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (Homf𝐶) = (Homf𝐷))
12 simprl 767 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
13 simprr 769 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶))
148, 9, 10, 11, 12, 13homfeqval 17434 . . . . . . 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 17447 . . . . . . . . . 10 (𝜑 → (Id‘𝐴) = (Id‘𝐵))
1918ad2antrr 722 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (Id‘𝐴) = (Id‘𝐵))
2019fveq1d 6794 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ((Id‘𝐴)‘𝑥) = ((Id‘𝐵)‘𝑥))
2120oveq1d 7310 . . . . . . 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 5168 . . . . . 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 7369 . . . . 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 4814 . . . 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 5166 . . 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 480 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐴)) → (Base‘𝐴) = (Base‘𝐵))
27 eqid 2733 . . . . . 6 (Base‘𝐴) = (Base‘𝐴)
28 eqid 2733 . . . . . 6 (Hom ‘𝐴) = (Hom ‘𝐴)
29 eqid 2733 . . . . . 6 (Hom ‘𝐵) = (Hom ‘𝐵)
301adantr 480 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (Homf𝐴) = (Homf𝐵))
31 simprl 767 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝐴))
32 simprr 769 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐴))
3327, 28, 29, 30, 31, 32homfeqval 17434 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
344ad2antrr 722 . . . . . 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 17447 . . . . . . . . 9 (𝜑 → (Id‘𝐶) = (Id‘𝐷))
3938ad3antrrr 726 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦)) ∧ 𝑧 ∈ (Base‘𝐶)) → (Id‘𝐶) = (Id‘𝐷))
4039fveq1d 6794 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦)) ∧ 𝑧 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑧) = ((Id‘𝐷)‘𝑧))
4140oveq2d 7311 . . . . . 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 5166 . . . . 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 5166 . . . 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 7369 . . 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 4814 . 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 2733 . . 3 (⟨𝐴, 𝐶⟩ curryF 𝐹) = (⟨𝐴, 𝐶⟩ curryF 𝐹)
47 curfpropd.f . . 3 (𝜑𝐹 ∈ ((𝐴 ×c 𝐶) Func 𝐸))
48 eqid 2733 . . 3 (Id‘𝐴) = (Id‘𝐴)
49 eqid 2733 . . 3 (Id‘𝐶) = (Id‘𝐶)
5046, 27, 16, 36, 47, 8, 9, 48, 28, 49curfval 17969 . 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 2733 . . 3 (⟨𝐵, 𝐷⟩ curryF 𝐹) = (⟨𝐵, 𝐷⟩ curryF 𝐹)
52 eqid 2733 . . 3 (Base‘𝐵) = (Base‘𝐵)
531, 15, 3, 35, 16, 17, 36, 37xpcpropd 17954 . . . . 5 (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))
5453oveq1d 7310 . . . 4 (𝜑 → ((𝐴 ×c 𝐶) Func 𝐸) = ((𝐵 ×c 𝐷) Func 𝐸))
5547, 54eleqtrd 2836 . . 3 (𝜑𝐹 ∈ ((𝐵 ×c 𝐷) Func 𝐸))
56 eqid 2733 . . 3 (Base‘𝐷) = (Base‘𝐷)
57 eqid 2733 . . 3 (Id‘𝐵) = (Id‘𝐵)
58 eqid 2733 . . 3 (Id‘𝐷) = (Id‘𝐷)
5951, 52, 17, 37, 55, 56, 10, 57, 29, 58curfval 17969 . 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 2783 1 (𝜑 → (⟨𝐴, 𝐶⟩ curryF 𝐹) = (⟨𝐵, 𝐷⟩ curryF 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2101  cop 4570  cmpt 5160  cfv 6447  (class class class)co 7295  cmpo 7297  1st c1st 7849  2nd c2nd 7850  Basecbs 16940  Hom chom 17001  Catccat 17401  Idccid 17402  Homf chomf 17403  compfccomf 17404   Func cfunc 17597   ×c cxpc 17913   curryF ccurf 17956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608  ax-cnex 10955  ax-resscn 10956  ax-1cn 10957  ax-icn 10958  ax-addcl 10959  ax-addrcl 10960  ax-mulcl 10961  ax-mulrcl 10962  ax-mulcom 10963  ax-addass 10964  ax-mulass 10965  ax-distr 10966  ax-i2m1 10967  ax-1ne0 10968  ax-1rid 10969  ax-rnegex 10970  ax-rrecex 10971  ax-cnre 10972  ax-pre-lttri 10973  ax-pre-lttrn 10974  ax-pre-ltadd 10975  ax-pre-mulgt0 10976
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3908  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-tr 5195  df-id 5491  df-eprel 5497  df-po 5505  df-so 5506  df-fr 5546  df-we 5548  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-riota 7252  df-ov 7298  df-oprab 7299  df-mpo 7300  df-om 7733  df-1st 7851  df-2nd 7852  df-frecs 8117  df-wrecs 8148  df-recs 8222  df-rdg 8261  df-1o 8317  df-er 8518  df-en 8754  df-dom 8755  df-sdom 8756  df-fin 8757  df-pnf 11039  df-mnf 11040  df-xr 11041  df-ltxr 11042  df-le 11043  df-sub 11235  df-neg 11236  df-nn 12002  df-2 12064  df-3 12065  df-4 12066  df-5 12067  df-6 12068  df-7 12069  df-8 12070  df-9 12071  df-n0 12262  df-z 12348  df-dec 12466  df-uz 12611  df-fz 13268  df-struct 16876  df-slot 16911  df-ndx 16923  df-base 16941  df-hom 17014  df-cco 17015  df-cat 17405  df-cid 17406  df-homf 17407  df-comf 17408  df-xpc 17917  df-curf 17960
This theorem is referenced by:  yonpropd  18014  oppcyon  18015
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