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Theorem fucco 17228
 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 2801 . . . 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 17216 . . . . . . . 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 498 . . . . . 6 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
10 funcrcl 17129 . . . . . 6 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
119, 10syl 17 . . . . 5 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1211simpld 498 . . . 4 (𝜑𝐶 ∈ Cat)
1311simprd 499 . . . 4 (𝜑𝐷 ∈ Cat)
14 fucco.x . . . 4 = (comp‘𝑄)
151, 2, 3, 4, 5, 12, 13, 14fuccofval 17225 . . 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 6664 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → (1st𝑣) ∈ V)
17 simprl 770 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → 𝑣 = ⟨𝐹, 𝐺⟩)
1817fveq2d 6653 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → (1st𝑣) = (1st ‘⟨𝐹, 𝐺⟩))
19 op1stg 7687 . . . . . . 7 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
208, 19syl 17 . . . . . 6 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
2120adantr 484 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
2218, 21eqtrd 2836 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → (1st𝑣) = 𝐹)
23 fvexd 6664 . . . . 5 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd𝑣) ∈ V)
2417adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → 𝑣 = ⟨𝐹, 𝐺⟩)
2524fveq2d 6653 . . . . . 6 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd𝑣) = (2nd ‘⟨𝐹, 𝐺⟩))
26 op2ndg 7688 . . . . . . . 8 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
278, 26syl 17 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
2827ad2antrr 725 . . . . . 6 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
2925, 28eqtrd 2836 . . . . 5 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd𝑣) = 𝐺)
30 simpr 488 . . . . . . 7 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
31 simprr 772 . . . . . . . 8 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → = 𝐻)
3231ad2antrr 725 . . . . . . 7 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → = 𝐻)
3330, 32oveq12d 7157 . . . . . 6 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔𝑁) = (𝐺𝑁𝐻))
34 simplr 768 . . . . . . 7 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
3534, 30oveq12d 7157 . . . . . 6 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑓𝑁𝑔) = (𝐹𝑁𝐺))
3634fveq2d 6653 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st𝑓) = (1st𝐹))
3736fveq1d 6651 . . . . . . . . . 10 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑥))
3830fveq2d 6653 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st𝑔) = (1st𝐺))
3938fveq1d 6651 . . . . . . . . . 10 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st𝑔)‘𝑥) = ((1st𝐺)‘𝑥))
4037, 39opeq12d 4776 . . . . . . . . 9 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)
4132fveq2d 6653 . . . . . . . . . 10 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st) = (1st𝐻))
4241fveq1d 6651 . . . . . . . . 9 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st)‘𝑥) = ((1st𝐻)‘𝑥))
4340, 42oveq12d 7157 . . . . . . . 8 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥)) = (⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥)))
4443oveqd 7156 . . . . . . 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 5129 . . . . . 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 7212 . . . . 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 3868 . . . 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 3868 . . 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 5560 . . . 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 17216 . . . . 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 499 . . 3 (𝜑𝐻 ∈ (𝐶 Func 𝐷))
55 ovex 7172 . . . . 5 (𝐺𝑁𝐻) ∈ V
56 ovex 7172 . . . . 5 (𝐹𝑁𝐺) ∈ V
5755, 56mpoex 7764 . . . 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 7285 . 2 (𝜑 → (⟨𝐹, 𝐺 𝐻) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)))))
60 simprl 770 . . . . 5 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → 𝑏 = 𝑆)
6160fveq1d 6651 . . . 4 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → (𝑏𝑥) = (𝑆𝑥))
62 simprr 772 . . . . 5 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → 𝑎 = 𝑅)
6362fveq1d 6651 . . . 4 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → (𝑎𝑥) = (𝑅𝑥))
6461, 63oveq12d 7157 . . 3 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)) = ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑅𝑥)))
6564mpteq2dv 5129 . 2 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥))) = (𝑥𝐴 ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑅𝑥))))
664fvexi 6663 . . . 4 𝐴 ∈ V
6766mptex 6967 . . 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 7285 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 399   = wceq 1538   ∈ wcel 2112  Vcvv 3444  ⦋csb 3831  ⟨cop 4534   ↦ cmpt 5113   × cxp 5521  ‘cfv 6328  (class class class)co 7139   ∈ cmpo 7141  1st c1st 7673  2nd c2nd 7674  Basecbs 16479  compcco 16573  Catccat 16931   Func cfunc 17120   Nat cnat 17207   FuncCat cfuc 17208 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12890  df-struct 16481  df-ndx 16482  df-slot 16483  df-base 16485  df-hom 16585  df-cco 16586  df-func 17124  df-nat 17209  df-fuc 17210 This theorem is referenced by:  fuccoval  17229  fuccocl  17230  fuclid  17232  fucrid  17233  fucass  17234  fucsect  17238  curfcl  17478
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