Theorem cicref 16813

Description: Isomorphism is reflexive. (Contributed by AV, 5-Apr-2020.)

Assertion

Ref	Expression
cicref	⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂( ≃𝑐 ‘𝐶)𝑂)

Proof of Theorem cicref

Step	Hyp	Ref	Expression
1		eqid 2825	. 2 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
2		eqid 2825	. 2 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		simpl 476	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
4		simpr 479	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
5		eqid 2825	. . 3 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
6		eqid 2825	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
7		eqid 2825	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
8		eqid 2825	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
9	2, 6, 7, 3, 4	catidcl 16695	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑂) ∈ (𝑂(Hom ‘𝐶)𝑂))
10	2, 6, 7, 3, 4, 8, 4, 9	catrid 16697	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → (((Id‘𝐶)‘𝑂)(⟨𝑂, 𝑂⟩(comp‘𝐶)𝑂)((Id‘𝐶)‘𝑂)) = ((Id‘𝐶)‘𝑂))
11		eqid 2825	. . . . . . 7 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
12	2, 6, 8, 7, 11, 3, 4, 4, 9, 9	issect2 16766	. . . . . 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)))
13	12, 12	anbi12d 626	. . . . 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))))
14	10, 10, 13	mpbir2and 706	. . . 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)))
15	2, 5, 3, 4, 4, 11	isinv 16772	. . . 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))))
16	14, 15	mpbird 249	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑂)(𝑂(Inv‘𝐶)𝑂)((Id‘𝐶)‘𝑂))
17	2, 5, 3, 4, 4, 1, 16	inviso1 16778	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑂) ∈ (𝑂(Iso‘𝐶)𝑂))
18	1, 2, 3, 4, 4, 17	brcici 16812	1 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂( ≃𝑐 ‘𝐶)𝑂)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ⟨cop 4403   class class class wbr 4873  ‘cfv 6123  (class class class)co 6905  Basecbs 16222  Hom chom 16316  compcco 16317  Catccat 16677  Idccid 16678  Sectcsect 16756  Invcinv 16757  Isociso 16758   ≃_𝑐 ccic 16807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-supp 7560  df-cat 16681  df-cid 16682  df-sect 16759  df-inv 16760  df-iso 16761  df-cic 16808

This theorem is referenced by:  cicer  16818