Theorem sectss 17932

Description: The section relation is a relation between morphisms from 𝑋 to 𝑌 and morphisms from 𝑌 to 𝑋. (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
sectss	⊢ (𝜑 → (𝑋𝑆𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))

Proof of Theorem sectss

Dummy variables 𝑓 𝑔 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		issect.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	1, 2, 3, 4, 5, 6, 7, 8	sectfval 17931	. 2 ⊢ (𝜑 → (𝑋𝑆𝑌) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))})
10		opabssxp 5742	. 2 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))} ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))
11	9, 10	eqsstrdi 3952	1 ⊢ (𝜑 → (𝑋𝑆𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895  ⟨cop 4834  {copab 5166   × cxp 5648  ‘cfv 6538  (class class class)co 7420  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:  isinv  17940  invss  17941  oppcsect2  17959  oppcinv  17960