Theorem sectffval 17659

Description: Value of the section operation. (Contributed by Mario Carneiro, 2-Jan-2017.) Removed redundant hypotheses. (Revised by Zhi Wang, 27-Oct-2025.)

Hypotheses

Ref	Expression
issect.b	⊢ 𝐵 = (Base‘𝐶)
issect.h	⊢ 𝐻 = (Hom ‘𝐶)
issect.o	⊢ · = (comp‘𝐶)
issect.i	⊢ 1 = (Id‘𝐶)
issect.s	⊢ 𝑆 = (Sect‘𝐶)
issect.c	⊢ (𝜑 → 𝐶 ∈ Cat)

Assertion

Ref	Expression
sectffval	⊢ (𝜑 → 𝑆 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥))}))

Distinct variable groups:   𝑓,𝑔,𝑥,𝑦, 1   𝑥,𝐵,𝑦   𝐶,𝑓,𝑔,𝑥,𝑦   𝜑,𝑓,𝑔,𝑥,𝑦   𝑓,𝐻,𝑔,𝑥,𝑦   · ,𝑓,𝑔,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑓,𝑔)   𝑆(𝑥,𝑦,𝑓,𝑔)

Proof of Theorem sectffval

Dummy variables 𝑐 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issect.s	. 2 ⊢ 𝑆 = (Sect‘𝐶)
2		issect.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		fveq2 6828	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4		issect.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
5	3, 4	eqtr4di 2786	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6		fvexd 6843	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) ∈ V)
7		fveq2 6828	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
8		issect.h	. . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐶)
9	7, 8	eqtr4di 2786	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
10		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ℎ = 𝐻)
11	10	oveqd 7369	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦))
12	11	eleq2d 2819	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑓 ∈ (𝑥ℎ𝑦) ↔ 𝑓 ∈ (𝑥𝐻𝑦)))
13	10	oveqd 7369	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑦ℎ𝑥) = (𝑦𝐻𝑥))
14	13	eleq2d 2819	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑔 ∈ (𝑦ℎ𝑥) ↔ 𝑔 ∈ (𝑦𝐻𝑥)))
15	12, 14	anbi12d 632	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ↔ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥))))
16		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → 𝑐 = 𝐶)
17	16	fveq2d 6832	. . . . . . . . . . . 12 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
18		issect.o	. . . . . . . . . . . 12 ⊢ · = (comp‘𝐶)
19	17, 18	eqtr4di 2786	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (comp‘𝑐) = · )
20	19	oveqd 7369	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥) = (⟨𝑥, 𝑦⟩ · 𝑥))
21	20	oveqd 7369	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓))
22	16	fveq2d 6832	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (Id‘𝑐) = (Id‘𝐶))
23		issect.i	. . . . . . . . . . 11 ⊢ 1 = (Id‘𝐶)
24	22, 23	eqtr4di 2786	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (Id‘𝑐) = 1 )
25	24	fveq1d 6830	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ((Id‘𝑐)‘𝑥) = ( 1 ‘𝑥))
26	21, 25	eqeq12d 2749	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥) ↔ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥)))
27	15, 26	anbi12d 632	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥)) ↔ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥))))
28	6, 9, 27	sbcied2 3782	. . . . . 6 ⊢ (𝑐 = 𝐶 → ([(Hom ‘𝑐) / ℎ]((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥)) ↔ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥))))
29	28	opabbidv 5159	. . . . 5 ⊢ (𝑐 = 𝐶 → {⟨𝑓, 𝑔⟩ ∣ [(Hom ‘𝑐) / ℎ]((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥))})
30	5, 5, 29	mpoeq123dv 7427	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ {⟨𝑓, 𝑔⟩ ∣ [(Hom ‘𝑐) / ℎ]((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥))}))
31		df-sect 17656	. . . 4 ⊢ Sect = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ {⟨𝑓, 𝑔⟩ ∣ [(Hom ‘𝑐) / ℎ]((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))}))
32	4	fvexi 6842	. . . . 5 ⊢ 𝐵 ∈ V
33	32, 32	mpoex 8017	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥))}) ∈ V
34	30, 31, 33	fvmpt 6935	. . 3 ⊢ (𝐶 ∈ Cat → (Sect‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥))}))
35	2, 34	syl 17	. 2 ⊢ (𝜑 → (Sect‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥))}))
36	1, 35	eqtrid 2780	1 ⊢ (𝜑 → 𝑆 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥))}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437  [wsbc 3737  ⟨cop 4581  {copab 5155  ‘cfv 6486  (class class class)co 7352   ∈ cmpo 7354  Basecbs 17122  Hom chom 17174  compcco 17175  Catccat 17572  Idccid 17573  Sectcsect 17653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-sect 17656

This theorem is referenced by:  sectfval  17660  sectrcl2  49148  sectfn  49154  sectpropdlem  49161