Theorem sectfn 49516

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 2737	. . 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 7393	. . . . 5 ⊢ (𝑥(Hom ‘𝐶)𝑦) ∈ V
3		ovex 7393	. . . . 5 ⊢ (𝑦(Hom ‘𝐶)𝑥) ∈ V
4	2, 3	xpex 7700	. . . 4 ⊢ ((𝑥(Hom ‘𝐶)𝑦) × (𝑦(Hom ‘𝐶)𝑥)) ∈ V
5		opabssxp 5716	. . . 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 5259	. . 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 8016	. 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 2737	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		eqid 2737	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
10		eqid 2737	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
11		eqid 2737	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
12		eqid 2737	. . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
13		id 22	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
14	8, 9, 10, 11, 12, 13	sectffval 17708	. . 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 6585	. 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 258	1 ⊢ (𝐶 ∈ Cat → (Sect‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4574  {copab 5148   × cxp 5622   Fn wfn 6487  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  Basecbs 17170  Hom chom 17222  compcco 17223  Catccat 17621  Idccid 17622  Sectcsect 17702

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-sect 17705

This theorem is referenced by:  sectpropdlem  49523