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Theorem cicref 16401
 Description: Isomorphism is reflexive. (Contributed by AV, 5-Apr-2020.)
Assertion
Ref Expression
cicref ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂( ≃𝑐𝐶)𝑂)

Proof of Theorem cicref
StepHypRef Expression
1 eqid 2621 . 2 (Iso‘𝐶) = (Iso‘𝐶)
2 eqid 2621 . 2 (Base‘𝐶) = (Base‘𝐶)
3 simpl 473 . 2 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
4 simpr 477 . 2 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
5 eqid 2621 . . 3 (Inv‘𝐶) = (Inv‘𝐶)
6 eqid 2621 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
7 eqid 2621 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
8 eqid 2621 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
92, 6, 7, 3, 4catidcl 16283 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑂) ∈ (𝑂(Hom ‘𝐶)𝑂))
102, 6, 7, 3, 4, 8, 4, 9catrid 16285 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → (((Id‘𝐶)‘𝑂)(⟨𝑂, 𝑂⟩(comp‘𝐶)𝑂)((Id‘𝐶)‘𝑂)) = ((Id‘𝐶)‘𝑂))
11 eqid 2621 . . . . . . 7 (Sect‘𝐶) = (Sect‘𝐶)
122, 6, 8, 7, 11, 3, 4, 4, 9, 9issect2 16354 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → (((Id‘𝐶)‘𝑂)(𝑂(Sect‘𝐶)𝑂)((Id‘𝐶)‘𝑂) ↔ (((Id‘𝐶)‘𝑂)(⟨𝑂, 𝑂⟩(comp‘𝐶)𝑂)((Id‘𝐶)‘𝑂)) = ((Id‘𝐶)‘𝑂)))
1312, 12anbi12d 746 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → ((((Id‘𝐶)‘𝑂)(𝑂(Sect‘𝐶)𝑂)((Id‘𝐶)‘𝑂) ∧ ((Id‘𝐶)‘𝑂)(𝑂(Sect‘𝐶)𝑂)((Id‘𝐶)‘𝑂)) ↔ ((((Id‘𝐶)‘𝑂)(⟨𝑂, 𝑂⟩(comp‘𝐶)𝑂)((Id‘𝐶)‘𝑂)) = ((Id‘𝐶)‘𝑂) ∧ (((Id‘𝐶)‘𝑂)(⟨𝑂, 𝑂⟩(comp‘𝐶)𝑂)((Id‘𝐶)‘𝑂)) = ((Id‘𝐶)‘𝑂))))
1410, 10, 13mpbir2and 956 . . . 4 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → (((Id‘𝐶)‘𝑂)(𝑂(Sect‘𝐶)𝑂)((Id‘𝐶)‘𝑂) ∧ ((Id‘𝐶)‘𝑂)(𝑂(Sect‘𝐶)𝑂)((Id‘𝐶)‘𝑂)))
152, 5, 3, 4, 4, 11isinv 16360 . . . 4 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → (((Id‘𝐶)‘𝑂)(𝑂(Inv‘𝐶)𝑂)((Id‘𝐶)‘𝑂) ↔ (((Id‘𝐶)‘𝑂)(𝑂(Sect‘𝐶)𝑂)((Id‘𝐶)‘𝑂) ∧ ((Id‘𝐶)‘𝑂)(𝑂(Sect‘𝐶)𝑂)((Id‘𝐶)‘𝑂))))
1614, 15mpbird 247 . . 3 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑂)(𝑂(Inv‘𝐶)𝑂)((Id‘𝐶)‘𝑂))
172, 5, 3, 4, 4, 1, 16inviso1 16366 . 2 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑂) ∈ (𝑂(Iso‘𝐶)𝑂))
181, 2, 3, 4, 4, 17brcici 16400 1 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂( ≃𝑐𝐶)𝑂)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ⟨cop 4161   class class class wbr 4623  ‘cfv 5857  (class class class)co 6615  Basecbs 15800  Hom chom 15892  compcco 15893  Catccat 16265  Idccid 16266  Sectcsect 16344  Invcinv 16345  Isociso 16346   ≃𝑐 ccic 16395 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-supp 7256  df-cat 16269  df-cid 16270  df-sect 16347  df-inv 16348  df-iso 16349  df-cic 16396 This theorem is referenced by:  cicer  16406
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