Theorem cicer 17851

Description: Isomorphism is an equivalence relation on objects of a category. Remark 3.16 in [Adamek] p. 29. (Contributed by AV, 5-Apr-2020.)

Assertion

Ref	Expression
cicer	⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) Er (Base‘𝐶))

Proof of Theorem cicer

Dummy variables 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopabv 5830	. . . . . 6 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)}
2	1	a1i 11	. . . . 5 ⊢ (𝐶 ∈ Cat → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)})
3		fveq2 6905	. . . . . . . . 9 ⊢ (𝑓 = ⟨𝑥, 𝑦⟩ → ((Iso‘𝐶)‘𝑓) = ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩))
4	3	neeq1d 2999	. . . . . . . 8 ⊢ (𝑓 = ⟨𝑥, 𝑦⟩ → (((Iso‘𝐶)‘𝑓) ≠ ∅ ↔ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅))
5	4	rabxp 5732	. . . . . . 7 ⊢ {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)}
6	5	a1i 11	. . . . . 6 ⊢ (𝐶 ∈ Cat → {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)})
7	6	releqd 5787	. . . . 5 ⊢ (𝐶 ∈ Cat → (Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅} ↔ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)}))
8	2, 7	mpbird 257	. . . 4 ⊢ (𝐶 ∈ Cat → Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
9		isofn 17820	. . . . . 6 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
10		fvex 6918	. . . . . . 7 ⊢ (Base‘𝐶) ∈ V
11		sqxpexg 7776	. . . . . . 7 ⊢ ((Base‘𝐶) ∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
12	10, 11	mp1i 13	. . . . . 6 ⊢ (𝐶 ∈ Cat → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
13		0ex 5306	. . . . . . 7 ⊢ ∅ ∈ V
14	13	a1i 11	. . . . . 6 ⊢ (𝐶 ∈ Cat → ∅ ∈ V)
15		suppvalfn 8194	. . . . . 6 ⊢ (((Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Base‘𝐶) × (Base‘𝐶)) ∈ V ∧ ∅ ∈ V) → ((Iso‘𝐶) supp ∅) = {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
16	9, 12, 14, 15	syl3anc 1372	. . . . 5 ⊢ (𝐶 ∈ Cat → ((Iso‘𝐶) supp ∅) = {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
17	16	releqd 5787	. . . 4 ⊢ (𝐶 ∈ Cat → (Rel ((Iso‘𝐶) supp ∅) ↔ Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅}))
18	8, 17	mpbird 257	. . 3 ⊢ (𝐶 ∈ Cat → Rel ((Iso‘𝐶) supp ∅))
19		cicfval 17842	. . . 4 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅))
20	19	releqd 5787	. . 3 ⊢ (𝐶 ∈ Cat → (Rel ( ≃𝑐 ‘𝐶) ↔ Rel ((Iso‘𝐶) supp ∅)))
21	18, 20	mpbird 257	. 2 ⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶))
22		cicsym 17849	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐 ‘𝐶)𝑦) → 𝑦( ≃𝑐 ‘𝐶)𝑥)
23		cictr 17850	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐 ‘𝐶)𝑦 ∧ 𝑦( ≃𝑐 ‘𝐶)𝑧) → 𝑥( ≃𝑐 ‘𝐶)𝑧)
24	23	3expb 1120	. 2 ⊢ ((𝐶 ∈ Cat ∧ (𝑥( ≃𝑐 ‘𝐶)𝑦 ∧ 𝑦( ≃𝑐 ‘𝐶)𝑧)) → 𝑥( ≃𝑐 ‘𝐶)𝑧)
25		cicref 17846	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥( ≃𝑐 ‘𝐶)𝑥)
26		ciclcl 17847	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐 ‘𝐶)𝑥) → 𝑥 ∈ (Base‘𝐶))
27	25, 26	impbida 800	. 2 ⊢ (𝐶 ∈ Cat → (𝑥 ∈ (Base‘𝐶) ↔ 𝑥( ≃𝑐 ‘𝐶)𝑥))
28	21, 22, 24, 27	iserd 8772	1 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) Er (Base‘𝐶))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  {crab 3435  Vcvv 3479  ∅c0 4332  ⟨cop 4631   class class class wbr 5142  {copab 5204   × cxp 5682  Rel wrel 5689   Fn wfn 6555  ‘cfv 6560  (class class class)co 7432   supp csupp 8186   Er wer 8743  Basecbs 17248  Catccat 17708  Isociso 17791   ≃_𝑐 ccic 17840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-supp 8187  df-er 8746  df-cat 17712  df-cid 17713  df-sect 17792  df-inv 17793  df-iso 17794  df-cic 17841

This theorem is referenced by: (None)