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Theorem fucsect 16984
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 16972 . . 3 (𝐶 Func 𝐷) = (Base‘𝑄)
3 fuciso.n . . . 4 𝑁 = (𝐶 Nat 𝐷)
41, 3fuchom 16973 . . 3 𝑁 = (Hom ‘𝑄)
5 eqid 2825 . . 3 (comp‘𝑄) = (comp‘𝑄)
6 eqid 2825 . . 3 (Id‘𝑄) = (Id‘𝑄)
7 fucsect.s . . 3 𝑆 = (Sect‘𝑄)
8 fuciso.f . . . . . 6 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
9 funcrcl 16875 . . . . . 6 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
108, 9syl 17 . . . . 5 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1110simpld 490 . . . 4 (𝜑𝐶 ∈ Cat)
1210simprd 491 . . . 4 (𝜑𝐷 ∈ Cat)
131, 11, 12fuccat 16982 . . 3 (𝜑𝑄 ∈ Cat)
14 fuciso.g . . 3 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
152, 4, 5, 6, 7, 13, 8, 14issect 16765 . 2 (𝜑 → (𝑈(𝐹𝑆𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹))))
16 ovex 6937 . . . . . . 7 ((𝑉𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)) ∈ V
1716rgenw 3133 . . . . . 6 𝑥𝐵 ((𝑉𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)) ∈ V
18 mpteqb 6546 . . . . . 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 2825 . . . . . . 7 (comp‘𝐷) = (comp‘𝐷)
22 simprl 789 . . . . . . 7 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝑈 ∈ (𝐹𝑁𝐺))
23 simprr 791 . . . . . . 7 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝑉 ∈ (𝐺𝑁𝐹))
241, 3, 20, 21, 5, 22, 23fucco 16974 . . . . . 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 2825 . . . . . . . 8 (Id‘𝐷) = (Id‘𝐷)
268adantr 474 . . . . . . . 8 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝐹 ∈ (𝐶 Func 𝐷))
271, 6, 25, 26fucid 16983 . . . . . . 7 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝑄)‘𝐹) = ((Id‘𝐷) ∘ (1st𝐹)))
2812adantr 474 . . . . . . . . . 10 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝐷 ∈ Cat)
29 eqid 2825 . . . . . . . . . . 11 (Base‘𝐷) = (Base‘𝐷)
3029, 25cidfn 16692 . . . . . . . . . 10 (𝐷 ∈ Cat → (Id‘𝐷) Fn (Base‘𝐷))
3128, 30syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (Id‘𝐷) Fn (Base‘𝐷))
32 dffn2 6280 . . . . . . . . 9 ((Id‘𝐷) Fn (Base‘𝐷) ↔ (Id‘𝐷):(Base‘𝐷)⟶V)
3331, 32sylib 210 . . . . . . . 8 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (Id‘𝐷):(Base‘𝐷)⟶V)
34 relfunc 16874 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
35 1st2ndbr 7479 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3634, 8, 35sylancr 583 . . . . . . . . . 10 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3720, 29, 36funcf1 16878 . . . . . . . . 9 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
3837adantr 474 . . . . . . . 8 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (1st𝐹):𝐵⟶(Base‘𝐷))
39 fcompt 6650 . . . . . . . 8 (((Id‘𝐷):(Base‘𝐷)⟶V ∧ (1st𝐹):𝐵⟶(Base‘𝐷)) → ((Id‘𝐷) ∘ (1st𝐹)) = (𝑥𝐵 ↦ ((Id‘𝐷)‘((1st𝐹)‘𝑥))))
4033, 38, 39syl2anc 581 . . . . . . 7 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝐷) ∘ (1st𝐹)) = (𝑥𝐵 ↦ ((Id‘𝐷)‘((1st𝐹)‘𝑥))))
4127, 40eqtrd 2861 . . . . . 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 2825 . . . . . . 7 (Hom ‘𝐷) = (Hom ‘𝐷)
44 fucsect.t . . . . . . 7 𝑇 = (Sect‘𝐷)
4528adantr 474 . . . . . . 7 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝐷 ∈ Cat)
4638ffvelrnda 6608 . . . . . . 7 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
47 1st2ndbr 7479 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
4834, 14, 47sylancr 583 . . . . . . . . . 10 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
4920, 29, 48funcf1 16878 . . . . . . . . 9 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝐷))
5049adantr 474 . . . . . . . 8 ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (1st𝐺):𝐵⟶(Base‘𝐷))
5150ffvelrnda 6608 . . . . . . 7 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
5222adantr 474 . . . . . . . . 9 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝑈 ∈ (𝐹𝑁𝐺))
533, 52nat1st2nd 16963 . . . . . . . 8 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝑈 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
54 simpr 479 . . . . . . . 8 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝑥𝐵)
553, 53, 20, 43, 54natcl 16965 . . . . . . 7 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → (𝑈𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)))
5623adantr 474 . . . . . . . . 9 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝑉 ∈ (𝐺𝑁𝐹))
573, 56nat1st2nd 16963 . . . . . . . 8 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → 𝑉 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐹), (2nd𝐹)⟩))
583, 57, 20, 43, 54natcl 16965 . . . . . . 7 (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥𝐵) → (𝑉𝑥) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
5929, 43, 21, 25, 44, 45, 46, 51, 55, 58issect2 16766 . . . . . 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 3194 . . . . 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 303 . . . 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 576 . . 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 1115 . . 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 1115 . . 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 306 . 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 271 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 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  wral 3117  Vcvv 3414  cop 4403   class class class wbr 4873  cmpt 4952  ccom 5346  Rel wrel 5347   Fn wfn 6118  wf 6119  cfv 6123  (class class class)co 6905  1st c1st 7426  2nd c2nd 7427  Basecbs 16222  Hom chom 16316  compcco 16317  Catccat 16677  Idccid 16678  Sectcsect 16756   Func cfunc 16866   Nat cnat 16953   FuncCat cfuc 16954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-map 8124  df-ixp 8176  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-5 11417  df-6 11418  df-7 11419  df-8 11420  df-9 11421  df-n0 11619  df-z 11705  df-dec 11822  df-uz 11969  df-fz 12620  df-struct 16224  df-ndx 16225  df-slot 16226  df-base 16228  df-hom 16329  df-cco 16330  df-cat 16681  df-cid 16682  df-sect 16759  df-func 16870  df-nat 16955  df-fuc 16956
This theorem is referenced by:  fucinv  16985
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