Theorem sectfn 49531

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 2741	. . 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 7392	. . . . 5 ⊢ (𝑥(Hom ‘𝐶)𝑦) ∈ V
3		ovex 7392	. . . . 5 ⊢ (𝑦(Hom ‘𝐶)𝑥) ∈ V
4	2, 3	xpex 7699	. . . 4 ⊢ ((𝑥(Hom ‘𝐶)𝑦) × (𝑦(Hom ‘𝐶)𝑥)) ∈ V
5		opabssxp 5712	. . . 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 5252	. . 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 8014	. 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 2741	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		eqid 2741	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
10		eqid 2741	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
11		eqid 2741	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
12		eqid 2741	. . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
13		id 22	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
14	8, 9, 10, 11, 12, 13	sectffval 17712	. . 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 6581	. 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 260	1 ⊢ (𝐶 ∈ Cat → (Sect‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ⟨cop 4563  {copab 5136   × cxp 5618   Fn wfn 6483  ‘cfv 6488  (class class class)co 7359   ∈ cmpo 7361  Basecbs 17174  Hom chom 17226  compcco 17227  Catccat 17625  Idccid 17626  Sectcsect 17706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7362  df-oprab 7363  df-mpo 7364  df-1st 7933  df-2nd 7934  df-sect 17709

This theorem is referenced by:  sectpropdlem  49538