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Theorem fucco 17982
Description: Value of the composition of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fucco.q 𝑄 = (𝐶 FuncCat 𝐷)
fucco.n 𝑁 = (𝐶 Nat 𝐷)
fucco.a 𝐴 = (Base‘𝐶)
fucco.o · = (comp‘𝐷)
fucco.x = (comp‘𝑄)
fucco.f (𝜑𝑅 ∈ (𝐹𝑁𝐺))
fucco.g (𝜑𝑆 ∈ (𝐺𝑁𝐻))
Assertion
Ref Expression
fucco (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) = (𝑥𝐴 ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑅𝑥))))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆   𝑥,𝐶   𝑥,𝐷   𝑥, ·   𝑥,𝐹   𝑥,𝐺   𝑥,𝐻
Allowed substitution hints:   𝑄(𝑥)   (𝑥)   𝑁(𝑥)

Proof of Theorem fucco
Dummy variables 𝑎 𝑏 𝑓 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fucco.q . . . 4 𝑄 = (𝐶 FuncCat 𝐷)
2 eqid 2725 . . . 4 (𝐶 Func 𝐷) = (𝐶 Func 𝐷)
3 fucco.n . . . 4 𝑁 = (𝐶 Nat 𝐷)
4 fucco.a . . . 4 𝐴 = (Base‘𝐶)
5 fucco.o . . . 4 · = (comp‘𝐷)
6 fucco.f . . . . . . . 8 (𝜑𝑅 ∈ (𝐹𝑁𝐺))
73natrcl 17968 . . . . . . . 8 (𝑅 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
86, 7syl 17 . . . . . . 7 (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
98simpld 493 . . . . . 6 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
10 funcrcl 17877 . . . . . 6 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
119, 10syl 17 . . . . 5 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1211simpld 493 . . . 4 (𝜑𝐶 ∈ Cat)
1311simprd 494 . . . 4 (𝜑𝐷 ∈ Cat)
14 fucco.x . . . 4 = (comp‘𝑄)
151, 2, 3, 4, 5, 12, 13, 14fuccofval 17978 . . 3 (𝜑 = (𝑣 ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)), ∈ (𝐶 Func 𝐷) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))))
16 fvexd 6915 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → (1st𝑣) ∈ V)
17 simprl 769 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → 𝑣 = ⟨𝐹, 𝐺⟩)
1817fveq2d 6904 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → (1st𝑣) = (1st ‘⟨𝐹, 𝐺⟩))
19 op1stg 8014 . . . . . . 7 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
208, 19syl 17 . . . . . 6 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
2120adantr 479 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
2218, 21eqtrd 2765 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → (1st𝑣) = 𝐹)
23 fvexd 6915 . . . . 5 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd𝑣) ∈ V)
2417adantr 479 . . . . . . 7 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → 𝑣 = ⟨𝐹, 𝐺⟩)
2524fveq2d 6904 . . . . . 6 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd𝑣) = (2nd ‘⟨𝐹, 𝐺⟩))
26 op2ndg 8015 . . . . . . . 8 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
278, 26syl 17 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
2827ad2antrr 724 . . . . . 6 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
2925, 28eqtrd 2765 . . . . 5 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd𝑣) = 𝐺)
30 simpr 483 . . . . . . 7 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
31 simprr 771 . . . . . . . 8 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → = 𝐻)
3231ad2antrr 724 . . . . . . 7 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → = 𝐻)
3330, 32oveq12d 7441 . . . . . 6 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔𝑁) = (𝐺𝑁𝐻))
34 simplr 767 . . . . . . 7 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
3534, 30oveq12d 7441 . . . . . 6 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑓𝑁𝑔) = (𝐹𝑁𝐺))
3634fveq2d 6904 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st𝑓) = (1st𝐹))
3736fveq1d 6902 . . . . . . . . . 10 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑥))
3830fveq2d 6904 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st𝑔) = (1st𝐺))
3938fveq1d 6902 . . . . . . . . . 10 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st𝑔)‘𝑥) = ((1st𝐺)‘𝑥))
4037, 39opeq12d 4886 . . . . . . . . 9 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)
4132fveq2d 6904 . . . . . . . . . 10 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st) = (1st𝐻))
4241fveq1d 6902 . . . . . . . . 9 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st)‘𝑥) = ((1st𝐻)‘𝑥))
4340, 42oveq12d 7441 . . . . . . . 8 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥)) = (⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥)))
4443oveqd 7440 . . . . . . 7 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)) = ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)))
4544mpteq2dv 5254 . . . . . 6 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))) = (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥))))
4633, 35, 45mpoeq123dv 7499 . . . . 5 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)))))
4723, 29, 46csbied2 3931 . . . 4 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)))))
4816, 22, 47csbied2 3931 . . 3 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)))))
49 opelxpi 5718 . . . 4 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → ⟨𝐹, 𝐺⟩ ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)))
508, 49syl 17 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)))
51 fucco.g . . . . 5 (𝜑𝑆 ∈ (𝐺𝑁𝐻))
523natrcl 17968 . . . . 5 (𝑆 ∈ (𝐺𝑁𝐻) → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)))
5351, 52syl 17 . . . 4 (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)))
5453simprd 494 . . 3 (𝜑𝐻 ∈ (𝐶 Func 𝐷))
55 ovex 7456 . . . . 5 (𝐺𝑁𝐻) ∈ V
56 ovex 7456 . . . . 5 (𝐹𝑁𝐺) ∈ V
5755, 56mpoex 8092 . . . 4 (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)))) ∈ V
5857a1i 11 . . 3 (𝜑 → (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)))) ∈ V)
5915, 48, 50, 54, 58ovmpod 7577 . 2 (𝜑 → (⟨𝐹, 𝐺 𝐻) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)))))
60 simprl 769 . . . . 5 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → 𝑏 = 𝑆)
6160fveq1d 6902 . . . 4 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → (𝑏𝑥) = (𝑆𝑥))
62 simprr 771 . . . . 5 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → 𝑎 = 𝑅)
6362fveq1d 6902 . . . 4 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → (𝑎𝑥) = (𝑅𝑥))
6461, 63oveq12d 7441 . . 3 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)) = ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑅𝑥)))
6564mpteq2dv 5254 . 2 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥))) = (𝑥𝐴 ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑅𝑥))))
664fvexi 6914 . . . 4 𝐴 ∈ V
6766mptex 7239 . . 3 (𝑥𝐴 ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑅𝑥))) ∈ V
6867a1i 11 . 2 (𝜑 → (𝑥𝐴 ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑅𝑥))) ∈ V)
6959, 65, 51, 6, 68ovmpod 7577 1 (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) = (𝑥𝐴 ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑅𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3461  csb 3891  cop 4638  cmpt 5235   × cxp 5679  cfv 6553  (class class class)co 7423  cmpo 7425  1st c1st 8000  2nd c2nd 8001  Basecbs 17208  compcco 17273  Catccat 17672   Func cfunc 17868   Nat cnat 17959   FuncCat cfuc 17960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745  ax-cnex 11210  ax-resscn 11211  ax-1cn 11212  ax-icn 11213  ax-addcl 11214  ax-addrcl 11215  ax-mulcl 11216  ax-mulrcl 11217  ax-mulcom 11218  ax-addass 11219  ax-mulass 11220  ax-distr 11221  ax-i2m1 11222  ax-1ne0 11223  ax-1rid 11224  ax-rnegex 11225  ax-rrecex 11226  ax-cnre 11227  ax-pre-lttri 11228  ax-pre-lttrn 11229  ax-pre-ltadd 11230  ax-pre-mulgt0 11231
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5579  df-eprel 5585  df-po 5593  df-so 5594  df-fr 5636  df-we 5638  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-pred 6311  df-ord 6378  df-on 6379  df-lim 6380  df-suc 6381  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7379  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7876  df-1st 8002  df-2nd 8003  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-er 8733  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11296  df-mnf 11297  df-xr 11298  df-ltxr 11299  df-le 11300  df-sub 11492  df-neg 11493  df-nn 12260  df-2 12322  df-3 12323  df-4 12324  df-5 12325  df-6 12326  df-7 12327  df-8 12328  df-9 12329  df-n0 12520  df-z 12606  df-dec 12725  df-uz 12870  df-fz 13534  df-struct 17144  df-slot 17179  df-ndx 17191  df-base 17209  df-hom 17285  df-cco 17286  df-func 17872  df-nat 17961  df-fuc 17962
This theorem is referenced by:  fuccoval  17983  fuccocl  17984  fuclid  17986  fucrid  17987  fucass  17988  fucsect  17992  curfcl  18252
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