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Theorem fucsect 17231
 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 17219 . . 3 (𝐶 Func 𝐷) = (Base‘𝑄)
3 fuciso.n . . . 4 𝑁 = (𝐶 Nat 𝐷)
41, 3fuchom 17220 . . 3 𝑁 = (Hom ‘𝑄)
5 eqid 2824 . . 3 (comp‘𝑄) = (comp‘𝑄)
6 eqid 2824 . . 3 (Id‘𝑄) = (Id‘𝑄)
7 fucsect.s . . 3 𝑆 = (Sect‘𝑄)
8 fuciso.f . . . . . 6 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
9 funcrcl 17122 . . . . . 6 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
108, 9syl 17 . . . . 5 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1110simpld 498 . . . 4 (𝜑𝐶 ∈ Cat)
1210simprd 499 . . . 4 (𝜑𝐷 ∈ Cat)
131, 11, 12fuccat 17229 . . 3 (𝜑𝑄 ∈ Cat)
14 fuciso.g . . 3 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
152, 4, 5, 6, 7, 13, 8, 14issect 17012 . 2 (𝜑 → (𝑈(𝐹𝑆𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹))))
16 ovex 7171 . . . . . . 7 ((𝑉𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)) ∈ V
1716rgenw 3144 . . . . . 6 𝑥𝐵 ((𝑉𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)) ∈ V
18 mpteqb 6768 . . . . . 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 2824 . . . . . . 7 (comp‘𝐷) = (comp‘𝐷)
22 simprl 770 . . . . . . 7 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝑈 ∈ (𝐹𝑁𝐺))
23 simprr 772 . . . . . . 7 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝑉 ∈ (𝐺𝑁𝐹))
241, 3, 20, 21, 5, 22, 23fucco 17221 . . . . . 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 2824 . . . . . . . 8 (Id‘𝐷) = (Id‘𝐷)
268adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝐹 ∈ (𝐶 Func 𝐷))
271, 6, 25, 26fucid 17230 . . . . . . 7 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝑄)‘𝐹) = ((Id‘𝐷) ∘ (1st𝐹)))
2812adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝐷 ∈ Cat)
29 eqid 2824 . . . . . . . . . . 11 (Base‘𝐷) = (Base‘𝐷)
3029, 25cidfn 16939 . . . . . . . . . 10 (𝐷 ∈ Cat → (Id‘𝐷) Fn (Base‘𝐷))
3128, 30syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (Id‘𝐷) Fn (Base‘𝐷))
32 dffn2 6497 . . . . . . . . 9 ((Id‘𝐷) Fn (Base‘𝐷) ↔ (Id‘𝐷):(Base‘𝐷)⟶V)
3331, 32sylib 221 . . . . . . . 8 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (Id‘𝐷):(Base‘𝐷)⟶V)
34 relfunc 17121 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
35 1st2ndbr 7724 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3634, 8, 35sylancr 590 . . . . . . . . . 10 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3720, 29, 36funcf1 17125 . . . . . . . . 9 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
3837adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (1st𝐹):𝐵⟶(Base‘𝐷))
39 fcompt 6876 . . . . . . . 8 (((Id‘𝐷):(Base‘𝐷)⟶V ∧ (1st𝐹):𝐵⟶(Base‘𝐷)) → ((Id‘𝐷) ∘ (1st𝐹)) = (𝑥𝐵 ↦ ((Id‘𝐷)‘((1st𝐹)‘𝑥))))
4033, 38, 39syl2anc 587 . . . . . . 7 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝐷) ∘ (1st𝐹)) = (𝑥𝐵 ↦ ((Id‘𝐷)‘((1st𝐹)‘𝑥))))
4127, 40eqtrd 2859 . . . . . 6 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝑄)‘𝐹) = (𝑥𝐵 ↦ ((Id‘𝐷)‘((1st𝐹)‘𝑥))))
4224, 41eqeq12d 2840 . . . . 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 2824 . . . . . . 7 (Hom ‘𝐷) = (Hom ‘𝐷)
44 fucsect.t . . . . . . 7 𝑇 = (Sect‘𝐷)
4528adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝐷 ∈ Cat)
4638ffvelrnda 6832 . . . . . . 7 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
47 1st2ndbr 7724 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
4834, 14, 47sylancr 590 . . . . . . . . . 10 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
4920, 29, 48funcf1 17125 . . . . . . . . 9 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝐷))
5049adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (1st𝐺):𝐵⟶(Base‘𝐷))
5150ffvelrnda 6832 . . . . . . 7 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
5222adantr 484 . . . . . . . . 9 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝑈 ∈ (𝐹𝑁𝐺))
533, 52nat1st2nd 17210 . . . . . . . 8 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝑈 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
54 simpr 488 . . . . . . . 8 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝑥𝐵)
553, 53, 20, 43, 54natcl 17212 . . . . . . 7 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → (𝑈𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)))
5623adantr 484 . . . . . . . . 9 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝑉 ∈ (𝐺𝑁𝐹))
573, 56nat1st2nd 17210 . . . . . . . 8 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝑉 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐹), (2nd𝐹)⟩))
583, 57, 20, 43, 54natcl 17212 . . . . . . 7 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → (𝑉𝑥) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
5929, 43, 21, 25, 44, 45, 46, 51, 55, 58issect2 17013 . . . . . 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 3190 . . . . 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 314 . . . 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 582 . . 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 1086 . . 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 1086 . . 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 317 . 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 282 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∀wral 3132  Vcvv 3479  ⟨cop 4554   class class class wbr 5047   ↦ cmpt 5127   ∘ ccom 5540  Rel wrel 5541   Fn wfn 6331  ⟶wf 6332  ‘cfv 6336  (class class class)co 7138  1st c1st 7670  2nd c2nd 7671  Basecbs 16472  Hom chom 16565  compcco 16566  Catccat 16924  Idccid 16925  Sectcsect 17003   Func cfunc 17113   Nat cnat 17200   FuncCat cfuc 17201 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 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-cnex 10578  ax-resscn 10579  ax-1cn 10580  ax-icn 10581  ax-addcl 10582  ax-addrcl 10583  ax-mulcl 10584  ax-mulrcl 10585  ax-mulcom 10586  ax-addass 10587  ax-mulass 10588  ax-distr 10589  ax-i2m1 10590  ax-1ne0 10591  ax-1rid 10592  ax-rnegex 10593  ax-rrecex 10594  ax-cnre 10595  ax-pre-lttri 10596  ax-pre-lttrn 10597  ax-pre-ltadd 10598  ax-pre-mulgt0 10599 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 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-pnf 10662  df-mnf 10663  df-xr 10664  df-ltxr 10665  df-le 10666  df-sub 10857  df-neg 10858  df-nn 11624  df-2 11686  df-3 11687  df-4 11688  df-5 11689  df-6 11690  df-7 11691  df-8 11692  df-9 11693  df-n0 11884  df-z 11968  df-dec 12085  df-uz 12230  df-fz 12884  df-struct 16474  df-ndx 16475  df-slot 16476  df-base 16478  df-hom 16578  df-cco 16579  df-cat 16928  df-cid 16929  df-sect 17006  df-func 17117  df-nat 17202  df-fuc 17203 This theorem is referenced by:  fucinv  17232
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