Theorem sectfval 16411

Description: Value of the section relation. (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
sectfval	⊢ (𝜑 → (𝑋𝑆𝑌) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))})

Distinct variable groups:   𝑓,𝑔, 1   𝐶,𝑓,𝑔   𝜑,𝑓,𝑔   𝑓,𝐻,𝑔   · ,𝑓,𝑔   𝑓,𝑋,𝑔   𝑓,𝑌,𝑔

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

Proof of Theorem sectfval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issect.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		issect.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
3		issect.o	. . 3 ⊢ · = (comp‘𝐶)
4		issect.i	. . 3 ⊢ 1 = (Id‘𝐶)
5		issect.s	. . 3 ⊢ 𝑆 = (Sect‘𝐶)
6		issect.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		issect.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	1, 2, 3, 4, 5, 6, 7, 7	sectffval 16410	. 2 ⊢ (𝜑 → 𝑆 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥))}))
9		simprl 794	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋)
10		simprr 796	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
11	9, 10	oveq12d 6668	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
12	11	eleq2d 2687	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑋𝐻𝑌)))
13	10, 9	oveq12d 6668	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑦𝐻𝑥) = (𝑌𝐻𝑋))
14	13	eleq2d 2687	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑔 ∈ (𝑦𝐻𝑥) ↔ 𝑔 ∈ (𝑌𝐻𝑋)))
15	12, 14	anbi12d 747	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
16	9, 10	opeq12d 4410	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
17	16, 9	oveq12d 6668	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (⟨𝑥, 𝑦⟩ · 𝑥) = (⟨𝑋, 𝑌⟩ · 𝑋))
18	17	oveqd 6667	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓))
19	9	fveq2d 6195	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ( 1 ‘𝑥) = ( 1 ‘𝑋))
20	18, 19	eqeq12d 2637	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥) ↔ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋)))
21	15, 20	anbi12d 747	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥)) ↔ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))))
22	21	opabbidv 4716	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))})
23		issect.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
24		ovex 6678	. . . . 5 ⊢ (𝑋𝐻𝑌) ∈ V
25		ovex 6678	. . . . 5 ⊢ (𝑌𝐻𝑋) ∈ V
26	24, 25	xpex 6962	. . . 4 ⊢ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)) ∈ V
27		opabssxp 5193	. . . 4 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))} ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))
28	26, 27	ssexi 4803	. . 3 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))} ∈ V
29	28	a1i 11	. 2 ⊢ (𝜑 → {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))} ∈ V)
30	8, 22, 7, 23, 29	ovmpt2d 6788	1 ⊢ (𝜑 → (𝑋𝑆𝑌) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))})

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183  {copab 4712   × cxp 5112  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Hom chom 15952  compcco 15953  Catccat 16325  Idccid 16326  Sectcsect 16404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-sect 16407

This theorem is referenced by:  sectss  16412  issect  16413  dfiso2  16432