Theorem sectffval 17379

Description: Value of the section operation. (Contributed by Mario Carneiro, 2-Jan-2017.)

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)
issect.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
issect.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

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   𝑓,𝑋,𝑔,𝑥,𝑦   𝑓,𝑌,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 6756	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4		issect.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
5	3, 4	eqtr4di 2797	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6		fvexd 6771	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) ∈ V)
7		fveq2 6756	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
8		issect.h	. . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐶)
9	7, 8	eqtr4di 2797	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
10		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ℎ = 𝐻)
11	10	oveqd 7272	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦))
12	11	eleq2d 2824	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑓 ∈ (𝑥ℎ𝑦) ↔ 𝑓 ∈ (𝑥𝐻𝑦)))
13	10	oveqd 7272	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑦ℎ𝑥) = (𝑦𝐻𝑥))
14	13	eleq2d 2824	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑔 ∈ (𝑦ℎ𝑥) ↔ 𝑔 ∈ (𝑦𝐻𝑥)))
15	12, 14	anbi12d 630	. . . . . . . 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 6760	. . . . . . . . . . . 12 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
18		issect.o	. . . . . . . . . . . 12 ⊢ · = (comp‘𝐶)
19	17, 18	eqtr4di 2797	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (comp‘𝑐) = · )
20	19	oveqd 7272	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥) = (⟨𝑥, 𝑦⟩ · 𝑥))
21	20	oveqd 7272	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓))
22	16	fveq2d 6760	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (Id‘𝑐) = (Id‘𝐶))
23		issect.i	. . . . . . . . . . 11 ⊢ 1 = (Id‘𝐶)
24	22, 23	eqtr4di 2797	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (Id‘𝑐) = 1 )
25	24	fveq1d 6758	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ((Id‘𝑐)‘𝑥) = ( 1 ‘𝑥))
26	21, 25	eqeq12d 2754	. . . . . . . 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 630	. . . . . . 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 3758	. . . . . 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 5136	. . . . 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 7328	. . . 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 17376	. . . 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 6770	. . . . 5 ⊢ 𝐵 ∈ V
33	32, 32	mpoex 7893	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥))}) ∈ V
34	30, 31, 33	fvmpt 6857	. . 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 2790	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 1539   ∈ wcel 2108  Vcvv 3422  [wsbc 3711  ⟨cop 4564  {copab 5132  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  Basecbs 16840  Hom chom 16899  compcco 16900  Catccat 17290  Idccid 17291  Sectcsect 17373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-sect 17376

This theorem is referenced by:  sectfval  17380