Theorem sectffval 17930

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 6885	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4		issect.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
5	3, 4	eqtr4di 2813	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6		fvexd 6900	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) ∈ V)
7		fveq2 6885	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
8		issect.h	. . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐶)
9	7, 8	eqtr4di 2813	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
10		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ℎ = 𝐻)
11	10	oveqd 7437	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦))
12	11	eleq2d 2846	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑓 ∈ (𝑥ℎ𝑦) ↔ 𝑓 ∈ (𝑥𝐻𝑦)))
13	10	oveqd 7437	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑦ℎ𝑥) = (𝑦𝐻𝑥))
14	13	eleq2d 2846	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑔 ∈ (𝑦ℎ𝑥) ↔ 𝑔 ∈ (𝑦𝐻𝑥)))
15	12, 14	anbi12d 644	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ↔ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥))))
16		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → 𝑐 = 𝐶)
17	16	fveq2d 6889	. . . . . . . . . . . 12 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
18		issect.o	. . . . . . . . . . . 12 ⊢ · = (comp‘𝐶)
19	17, 18	eqtr4di 2813	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (comp‘𝑐) = · )
20	19	oveqd 7437	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥) = (⟨𝑥, 𝑦⟩ · 𝑥))
21	20	oveqd 7437	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓))
22	16	fveq2d 6889	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (Id‘𝑐) = (Id‘𝐶))
23		issect.i	. . . . . . . . . . 11 ⊢ 1 = (Id‘𝐶)
24	22, 23	eqtr4di 2813	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (Id‘𝑐) = 1 )
25	24	fveq1d 6887	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ((Id‘𝑐)‘𝑥) = ( 1 ‘𝑥))
26	21, 25	eqeq12d 2776	. . . . . . . 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 644	. . . . . . 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 5170	. . . . 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 7495	. . . 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 17927	. . . 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 6899	. . . . 5 ⊢ 𝐵 ∈ V
33	32, 32	mpoex 8092	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥))}) ∈ V
34	30, 31, 33	fvmpt 6993	. . 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 18	. 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 2807	1 ⊢ (𝜑 → 𝑆 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥))}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450  [wsbc 3738  ⟨cop 4834  {copab 5166  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  Basecbs 17392  Hom chom 17444  compcco 17445  Catccat 17843  Idccid 17844  Sectcsect 17924

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-sect 17927

This theorem is used by:  sectfval  17931  sectrcl2  50230  sectfn  50236  sectpropdlem  50243