Theorem cicer 17853

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 5799	. . . . . 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 6871	. . . . . . . . 9 ⊢ (𝑓 = ⟨𝑥, 𝑦⟩ → ((Iso‘𝐶)‘𝑓) = ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩))
4	3	neeq1d 3019	. . . . . . . 8 ⊢ (𝑓 = ⟨𝑥, 𝑦⟩ → (((Iso‘𝐶)‘𝑓) ≠ ∅ ↔ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅))
5	4	rabxp 5700	. . . . . . 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 5756	. . . . 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 260	. . . 4 ⊢ (𝐶 ∈ Cat → Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
9		isofn 17822	. . . . . 6 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
10		fvex 6884	. . . . . . 7 ⊢ (Base‘𝐶) ∈ V
11		sqxpexg 7742	. . . . . . 7 ⊢ ((Base‘𝐶) ∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
12	10, 11	mp1i 14	. . . . . 6 ⊢ (𝐶 ∈ Cat → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
13		0ex 5262	. . . . . . 7 ⊢ ∅ ∈ V
14	13	a1i 11	. . . . . 6 ⊢ (𝐶 ∈ Cat → ∅ ∈ V)
15		suppvalfn 8152	. . . . . 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 1394	. . . . 5 ⊢ (𝐶 ∈ Cat → ((Iso‘𝐶) supp ∅) = {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
17	16	releqd 5756	. . . 4 ⊢ (𝐶 ∈ Cat → (Rel ((Iso‘𝐶) supp ∅) ↔ Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅}))
18	8, 17	mpbird 260	. . 3 ⊢ (𝐶 ∈ Cat → Rel ((Iso‘𝐶) supp ∅))
19		cicfval 17844	. . . 4 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅))
20	19	releqd 5756	. . 3 ⊢ (𝐶 ∈ Cat → (Rel ( ≃𝑐 ‘𝐶) ↔ Rel ((Iso‘𝐶) supp ∅)))
21	18, 20	mpbird 260	. 2 ⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶))
22		cicsym 17851	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐 ‘𝐶)𝑦) → 𝑦( ≃𝑐 ‘𝐶)𝑥)
23		cictr 17852	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐 ‘𝐶)𝑦 ∧ 𝑦( ≃𝑐 ‘𝐶)𝑧) → 𝑥( ≃𝑐 ‘𝐶)𝑧)
24	23	3expb 1136	. 2 ⊢ ((𝐶 ∈ Cat ∧ (𝑥( ≃𝑐 ‘𝐶)𝑦 ∧ 𝑦( ≃𝑐 ‘𝐶)𝑧)) → 𝑥( ≃𝑐 ‘𝐶)𝑧)
25		cicref 17848	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥( ≃𝑐 ‘𝐶)𝑥)
26		ciclcl 17849	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐 ‘𝐶)𝑥) → 𝑥 ∈ (Base‘𝐶))
27	25, 26	impbida 812	. 2 ⊢ (𝐶 ∈ Cat → (𝑥 ∈ (Base‘𝐶) ↔ 𝑥( ≃𝑐 ‘𝐶)𝑥))
28	21, 22, 24, 27	iserd 8709	1 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) Er (Base‘𝐶))

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  {crab 3417  Vcvv 3457  ∅c0 4288  ⟨cop 4591   class class class wbr 5105  {copab 5167   × cxp 5650  Rel wrel 5657   Fn wfn 6520  ‘cfv 6525  (class class class)co 7400   supp csupp 8144   Er wer 8679  Basecbs 17259  Catccat 17710  Isociso 17793   ≃_𝑐 ccic 17842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-supp 8145  df-er 8682  df-cat 17714  df-cid 17715  df-sect 17794  df-inv 17795  df-iso 17796  df-cic 17843

This theorem is referenced by: (None)