Theorem issect 17816

Description: The property "𝐹 is a section of 𝐺". (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
issect	⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋))))

Proof of Theorem issect

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issect.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		issect.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
3		issect.o	. . . 4 ⊢ · = (comp‘𝐶)
4		issect.i	. . . 4 ⊢ 1 = (Id‘𝐶)
5		issect.s	. . . 4 ⊢ 𝑆 = (Sect‘𝐶)
6		issect.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		issect.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		issect.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	1, 2, 3, 4, 5, 6, 7, 8	sectfval 17814	. . 3 ⊢ (𝜑 → (𝑋𝑆𝑌) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))})
10	9	breqd 5119	. 2 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ 𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))}𝐺))
11		oveq12 7421	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹))
12	11	ancoms 463	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹))
13	12	eqeq1d 2764	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋) ↔ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)))
14		eqid 2762	. . . 4 ⊢ {⟨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))}
15	13, 14	brab2a 5753	. . 3 ⊢ (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))}𝐺 ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)))
16		df-3an 1104	. . 3 ⊢ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)) ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)))
17	15, 16	bitr4i 281	. 2 ⊢ (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))}𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)))
18	10, 17	bitrdi 290	1 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ⟨cop 4594   class class class wbr 5108  {copab 5172  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  Hom chom 17327  compcco 17328  Catccat 17726  Idccid 17727  Sectcsect 17807

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-sect 17810

This theorem is used by:  issect2  17817  sectcan  17818  sectco  17819  oppcsect  17841  sectmon  17845  monsect  17846  funcsect  17935  fucsect  18038  invfuc  18040  setcsect  18152  catciso  18174  rngcsect  20746  ringcsect  20780  rngcsectALTV  49068  ringcsectALTV  49102  catcsect  50204  thincsect  50273