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Theorem fucsect 17230
Description: Two natural transformations are in a section iff all the components are in a section relation. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
fuciso.q 𝑄 = (𝐶 FuncCat 𝐷)
fuciso.b 𝐵 = (Base‘𝐶)
fuciso.n 𝑁 = (𝐶 Nat 𝐷)
fuciso.f (𝜑𝐹 ∈ (𝐶 Func 𝐷))
fuciso.g (𝜑𝐺 ∈ (𝐶 Func 𝐷))
fucsect.s 𝑆 = (Sect‘𝑄)
fucsect.t 𝑇 = (Sect‘𝐷)
Assertion
Ref Expression
fucsect (𝜑 → (𝑈(𝐹𝑆𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝑇((1st𝐺)‘𝑥))(𝑉𝑥))))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷   𝑥,𝐹   𝑥,𝐺   𝑥,𝑁   𝑥,𝑉   𝜑,𝑥   𝑥,𝑄   𝑥,𝑈
Allowed substitution hints:   𝑆(𝑥)   𝑇(𝑥)

Proof of Theorem fucsect
StepHypRef Expression
1 fuciso.q . . . 4 𝑄 = (𝐶 FuncCat 𝐷)
21fucbas 17218 . . 3 (𝐶 Func 𝐷) = (Base‘𝑄)
3 fuciso.n . . . 4 𝑁 = (𝐶 Nat 𝐷)
41, 3fuchom 17219 . . 3 𝑁 = (Hom ‘𝑄)
5 eqid 2818 . . 3 (comp‘𝑄) = (comp‘𝑄)
6 eqid 2818 . . 3 (Id‘𝑄) = (Id‘𝑄)
7 fucsect.s . . 3 𝑆 = (Sect‘𝑄)
8 fuciso.f . . . . . 6 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
9 funcrcl 17121 . . . . . 6 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
108, 9syl 17 . . . . 5 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1110simpld 495 . . . 4 (𝜑𝐶 ∈ Cat)
1210simprd 496 . . . 4 (𝜑𝐷 ∈ Cat)
131, 11, 12fuccat 17228 . . 3 (𝜑𝑄 ∈ Cat)
14 fuciso.g . . 3 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
152, 4, 5, 6, 7, 13, 8, 14issect 17011 . 2 (𝜑 → (𝑈(𝐹𝑆𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹))))
16 ovex 7178 . . . . . . 7 ((𝑉𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)) ∈ V
1716rgenw 3147 . . . . . 6 𝑥𝐵 ((𝑉𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)) ∈ V
18 mpteqb 6779 . . . . . 6 (∀𝑥𝐵 ((𝑉𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)) ∈ V → ((𝑥𝐵 ↦ ((𝑉𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))) = (𝑥𝐵 ↦ ((Id‘𝐷)‘((1st𝐹)‘𝑥))) ↔ ∀𝑥𝐵 ((𝑉𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)) = ((Id‘𝐷)‘((1st𝐹)‘𝑥))))
1917, 18mp1i 13 . . . . 5 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((𝑥𝐵 ↦ ((𝑉𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))) = (𝑥𝐵 ↦ ((Id‘𝐷)‘((1st𝐹)‘𝑥))) ↔ ∀𝑥𝐵 ((𝑉𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)) = ((Id‘𝐷)‘((1st𝐹)‘𝑥))))
20 fuciso.b . . . . . . 7 𝐵 = (Base‘𝐶)
21 eqid 2818 . . . . . . 7 (comp‘𝐷) = (comp‘𝐷)
22 simprl 767 . . . . . . 7 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝑈 ∈ (𝐹𝑁𝐺))
23 simprr 769 . . . . . . 7 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝑉 ∈ (𝐺𝑁𝐹))
241, 3, 20, 21, 5, 22, 23fucco 17220 . . . . . 6 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = (𝑥𝐵 ↦ ((𝑉𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))))
25 eqid 2818 . . . . . . . 8 (Id‘𝐷) = (Id‘𝐷)
268adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝐹 ∈ (𝐶 Func 𝐷))
271, 6, 25, 26fucid 17229 . . . . . . 7 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝑄)‘𝐹) = ((Id‘𝐷) ∘ (1st𝐹)))
2812adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝐷 ∈ Cat)
29 eqid 2818 . . . . . . . . . . 11 (Base‘𝐷) = (Base‘𝐷)
3029, 25cidfn 16938 . . . . . . . . . 10 (𝐷 ∈ Cat → (Id‘𝐷) Fn (Base‘𝐷))
3128, 30syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (Id‘𝐷) Fn (Base‘𝐷))
32 dffn2 6509 . . . . . . . . 9 ((Id‘𝐷) Fn (Base‘𝐷) ↔ (Id‘𝐷):(Base‘𝐷)⟶V)
3331, 32sylib 219 . . . . . . . 8 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (Id‘𝐷):(Base‘𝐷)⟶V)
34 relfunc 17120 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
35 1st2ndbr 7730 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3634, 8, 35sylancr 587 . . . . . . . . . 10 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3720, 29, 36funcf1 17124 . . . . . . . . 9 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
3837adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (1st𝐹):𝐵⟶(Base‘𝐷))
39 fcompt 6887 . . . . . . . 8 (((Id‘𝐷):(Base‘𝐷)⟶V ∧ (1st𝐹):𝐵⟶(Base‘𝐷)) → ((Id‘𝐷) ∘ (1st𝐹)) = (𝑥𝐵 ↦ ((Id‘𝐷)‘((1st𝐹)‘𝑥))))
4033, 38, 39syl2anc 584 . . . . . . 7 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝐷) ∘ (1st𝐹)) = (𝑥𝐵 ↦ ((Id‘𝐷)‘((1st𝐹)‘𝑥))))
4127, 40eqtrd 2853 . . . . . 6 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝑄)‘𝐹) = (𝑥𝐵 ↦ ((Id‘𝐷)‘((1st𝐹)‘𝑥))))
4224, 41eqeq12d 2834 . . . . 5 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹) ↔ (𝑥𝐵 ↦ ((𝑉𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))) = (𝑥𝐵 ↦ ((Id‘𝐷)‘((1st𝐹)‘𝑥)))))
43 eqid 2818 . . . . . . 7 (Hom ‘𝐷) = (Hom ‘𝐷)
44 fucsect.t . . . . . . 7 𝑇 = (Sect‘𝐷)
4528adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝐷 ∈ Cat)
4638ffvelrnda 6843 . . . . . . 7 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
47 1st2ndbr 7730 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
4834, 14, 47sylancr 587 . . . . . . . . . 10 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
4920, 29, 48funcf1 17124 . . . . . . . . 9 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝐷))
5049adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (1st𝐺):𝐵⟶(Base‘𝐷))
5150ffvelrnda 6843 . . . . . . 7 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
5222adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝑈 ∈ (𝐹𝑁𝐺))
533, 52nat1st2nd 17209 . . . . . . . 8 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝑈 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
54 simpr 485 . . . . . . . 8 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝑥𝐵)
553, 53, 20, 43, 54natcl 17211 . . . . . . 7 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → (𝑈𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)))
5623adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝑉 ∈ (𝐺𝑁𝐹))
573, 56nat1st2nd 17209 . . . . . . . 8 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝑉 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐹), (2nd𝐹)⟩))
583, 57, 20, 43, 54natcl 17211 . . . . . . 7 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → (𝑉𝑥) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
5929, 43, 21, 25, 44, 45, 46, 51, 55, 58issect2 17012 . . . . . 6 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → ((𝑈𝑥)(((1st𝐹)‘𝑥)𝑇((1st𝐺)‘𝑥))(𝑉𝑥) ↔ ((𝑉𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)) = ((Id‘𝐷)‘((1st𝐹)‘𝑥))))
6059ralbidva 3193 . . . . 5 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝑇((1st𝐺)‘𝑥))(𝑉𝑥) ↔ ∀𝑥𝐵 ((𝑉𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)) = ((Id‘𝐷)‘((1st𝐹)‘𝑥))))
6119, 42, 603bitr4d 312 . . . 4 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹) ↔ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝑇((1st𝐺)‘𝑥))(𝑉𝑥)))
6261pm5.32da 579 . . 3 (𝜑 → (((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝑇((1st𝐺)‘𝑥))(𝑉𝑥))))
63 df-3an 1081 . . 3 ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹)))
64 df-3an 1081 . . 3 ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝑇((1st𝐺)‘𝑥))(𝑉𝑥)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝑇((1st𝐺)‘𝑥))(𝑉𝑥)))
6562, 63, 643bitr4g 315 . 2 (𝜑 → ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹)) ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝑇((1st𝐺)‘𝑥))(𝑉𝑥))))
6615, 65bitrd 280 1 (𝜑 → (𝑈(𝐹𝑆𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝑇((1st𝐺)‘𝑥))(𝑉𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cop 4563   class class class wbr 5057  cmpt 5137  ccom 5552  Rel wrel 5553   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  Basecbs 16471  Hom chom 16564  compcco 16565  Catccat 16923  Idccid 16924  Sectcsect 17002   Func cfunc 17112   Nat cnat 17199   FuncCat cfuc 17200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12881  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-hom 16577  df-cco 16578  df-cat 16927  df-cid 16928  df-sect 17005  df-func 17116  df-nat 17201  df-fuc 17202
This theorem is referenced by:  fucinv  17231
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