Theorem sectfn 49687

Description: The function value of the function returning the sections of a category is a function over the Cartesian square of the base set of the category. (Contributed by Zhi Wang, 27-Oct-2025.)

Assertion

Ref	Expression
sectfn	⊢ (𝐶 ∈ Cat → (Sect‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))

Proof of Theorem sectfn

Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2769	. . 3 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑥)𝑓) = ((Id‘𝐶)‘𝑥))}) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑥)𝑓) = ((Id‘𝐶)‘𝑥))})
2		ovex 7441	. . . . 5 ⊢ (𝑥(Hom ‘𝐶)𝑦) ∈ V
3		ovex 7441	. . . . 5 ⊢ (𝑦(Hom ‘𝐶)𝑥) ∈ V
4	2, 3	xpex 7748	. . . 4 ⊢ ((𝑥(Hom ‘𝐶)𝑦) × (𝑦(Hom ‘𝐶)𝑥)) ∈ V
5		opabssxp 5751	. . . 4 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑥)𝑓) = ((Id‘𝐶)‘𝑥))} ⊆ ((𝑥(Hom ‘𝐶)𝑦) × (𝑦(Hom ‘𝐶)𝑥))
6	4, 5	ssexi 5290	. . 3 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑥)𝑓) = ((Id‘𝐶)‘𝑥))} ∈ V
7	1, 6	fnmpoi 8063	. 2 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑥)𝑓) = ((Id‘𝐶)‘𝑥))}) Fn ((Base‘𝐶) × (Base‘𝐶))
8		eqid 2769	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		eqid 2769	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
10		eqid 2769	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
11		eqid 2769	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
12		eqid 2769	. . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
13		id 23	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
14	8, 9, 10, 11, 12, 13	sectffval 17803	. . 3 ⊢ (𝐶 ∈ Cat → (Sect‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑥)𝑓) = ((Id‘𝐶)‘𝑥))}))
15	14	fneq1d 6626	. 2 ⊢ (𝐶 ∈ Cat → ((Sect‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑥)𝑓) = ((Id‘𝐶)‘𝑥))}) Fn ((Base‘𝐶) × (Base‘𝐶))))
16	7, 15	mpbiri 261	1 ⊢ (𝐶 ∈ Cat → (Sect‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ⟨cop 4597  {copab 5174   × cxp 5657   Fn wfn 6529  ‘cfv 6534  (class class class)co 7408   ∈ cmpo 7410  Basecbs 17265  Hom chom 17317  compcco 17318  Catccat 17716  Idccid 17717  Sectcsect 17797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-sect 17800

This theorem is referenced by:  sectpropdlem  49694