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Theorem invss 17751
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 2728 . . . 4 (Sect‘𝐶) = (Sect‘𝐶)
71, 2, 3, 4, 5, 6invfval 17749 . . 3 (𝜑 → (𝑋𝑁𝑌) = ((𝑋(Sect‘𝐶)𝑌) ∩ (𝑌(Sect‘𝐶)𝑋)))
8 inss1 4231 . . 3 ((𝑋(Sect‘𝐶)𝑌) ∩ (𝑌(Sect‘𝐶)𝑋)) ⊆ (𝑋(Sect‘𝐶)𝑌)
97, 8eqsstrdi 4036 . 2 (𝜑 → (𝑋𝑁𝑌) ⊆ (𝑋(Sect‘𝐶)𝑌))
10 invss.h . . 3 𝐻 = (Hom ‘𝐶)
11 eqid 2728 . . 3 (comp‘𝐶) = (comp‘𝐶)
12 eqid 2728 . . 3 (Id‘𝐶) = (Id‘𝐶)
131, 10, 11, 12, 6, 3, 4, 5sectss 17742 . 2 (𝜑 → (𝑋(Sect‘𝐶)𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))
149, 13sstrd 3992 1 (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cin 3948  wss 3949   × cxp 5680  ccnv 5681  cfv 6553  (class class class)co 7426  Basecbs 17187  Hom chom 17251  compcco 17252  Catccat 17651  Idccid 17652  Sectcsect 17734  Invcinv 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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-sect 17737  df-inv 17738
This theorem is referenced by:  invsym2  17753  invfun  17754  isohom  17766  invfuc  17973
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