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Theorem invss 16629
Description: The inverse relation is a relation between morphisms 𝐹:𝑋𝑌 and their inverses 𝐺:𝑌𝑋. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
invfval.b 𝐵 = (Base‘𝐶)
invfval.n 𝑁 = (Inv‘𝐶)
invfval.c (𝜑𝐶 ∈ Cat)
invfval.x (𝜑𝑋𝐵)
invfval.y (𝜑𝑌𝐵)
invss.h 𝐻 = (Hom ‘𝐶)
Assertion
Ref Expression
invss (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))

Proof of Theorem invss
StepHypRef Expression
1 invfval.b . . . 4 𝐵 = (Base‘𝐶)
2 invfval.n . . . 4 𝑁 = (Inv‘𝐶)
3 invfval.c . . . 4 (𝜑𝐶 ∈ Cat)
4 invfval.x . . . 4 (𝜑𝑋𝐵)
5 invfval.y . . . 4 (𝜑𝑌𝐵)
6 eqid 2771 . . . 4 (Sect‘𝐶) = (Sect‘𝐶)
71, 2, 3, 4, 5, 6invfval 16627 . . 3 (𝜑 → (𝑋𝑁𝑌) = ((𝑋(Sect‘𝐶)𝑌) ∩ (𝑌(Sect‘𝐶)𝑋)))
8 inss1 3982 . . 3 ((𝑋(Sect‘𝐶)𝑌) ∩ (𝑌(Sect‘𝐶)𝑋)) ⊆ (𝑋(Sect‘𝐶)𝑌)
97, 8syl6eqss 3805 . 2 (𝜑 → (𝑋𝑁𝑌) ⊆ (𝑋(Sect‘𝐶)𝑌))
10 invss.h . . 3 𝐻 = (Hom ‘𝐶)
11 eqid 2771 . . 3 (comp‘𝐶) = (comp‘𝐶)
12 eqid 2771 . . 3 (Id‘𝐶) = (Id‘𝐶)
131, 10, 11, 12, 6, 3, 4, 5sectss 16620 . 2 (𝜑 → (𝑋(Sect‘𝐶)𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))
149, 13sstrd 3763 1 (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cin 3723  wss 3724   × cxp 5248  ccnv 5249  cfv 6032  (class class class)co 6794  Basecbs 16065  Hom chom 16161  compcco 16162  Catccat 16533  Idccid 16534  Sectcsect 16612  Invcinv 16613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-1st 7316  df-2nd 7317  df-sect 16615  df-inv 16616
This theorem is referenced by:  invsym2  16631  invfun  16632  isohom  16644  invfuc  16842
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