MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sectss Structured version   Visualization version   GIF version

Theorem sectss 17743
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 (𝜑𝐶 ∈ Cat)
issect.x (𝜑𝑋𝐵)
issect.y (𝜑𝑌𝐵)
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.
StepHypRef 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 (𝜑𝐶 ∈ Cat)
7 issect.x . . 3 (𝜑𝑋𝐵)
8 issect.y . . 3 (𝜑𝑌𝐵)
91, 2, 3, 4, 5, 6, 7, 8sectfval 17742 . 2 (𝜑 → (𝑋𝑆𝑌) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))})
10 opabssxp 5770 . 2 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))} ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))
119, 10eqsstrdi 4031 1 (𝜑 → (𝑋𝑆𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wss 3944  cop 4636  {copab 5211   × cxp 5676  cfv 6549  (class class class)co 7419  Basecbs 17188  Hom chom 17252  compcco 17253  Catccat 17652  Idccid 17653  Sectcsect 17735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-sect 17738
This theorem is referenced by:  isinv  17751  invss  17752  oppcsect2  17770  oppcinv  17771
  Copyright terms: Public domain W3C validator