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Theorem sectss 17127
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 17126 . 2 (𝜑 → (𝑋𝑆𝑌) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))})
10 opabssxp 5614 . 2 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))} ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))
119, 10eqsstrdi 3931 1 (𝜑 → (𝑋𝑆𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  wss 3843  cop 4522  {copab 5092   × cxp 5523  cfv 6339  (class class class)co 7170  Basecbs 16586  Hom chom 16679  compcco 16680  Catccat 17038  Idccid 17039  Sectcsect 17119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715  df-sect 17122
This theorem is referenced by:  isinv  17135  invss  17136  oppcsect2  17154  oppcinv  17155
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