Theorem cicer 17507

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 5726	. . . . . 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 6768	. . . . . . . . 9 ⊢ (𝑓 = ⟨𝑥, 𝑦⟩ → ((Iso‘𝐶)‘𝑓) = ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩))
4	3	neeq1d 3003	. . . . . . . 8 ⊢ (𝑓 = ⟨𝑥, 𝑦⟩ → (((Iso‘𝐶)‘𝑓) ≠ ∅ ↔ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅))
5	4	rabxp 5632	. . . . . . 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 5685	. . . . 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 256	. . . 4 ⊢ (𝐶 ∈ Cat → Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
9		isofn 17476	. . . . . 6 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
10		fvex 6781	. . . . . . 7 ⊢ (Base‘𝐶) ∈ V
11		sqxpexg 7597	. . . . . . 7 ⊢ ((Base‘𝐶) ∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
12	10, 11	mp1i 13	. . . . . 6 ⊢ (𝐶 ∈ Cat → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
13		0ex 5231	. . . . . . 7 ⊢ ∅ ∈ V
14	13	a1i 11	. . . . . 6 ⊢ (𝐶 ∈ Cat → ∅ ∈ V)
15		suppvalfn 7974	. . . . . 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 1370	. . . . 5 ⊢ (𝐶 ∈ Cat → ((Iso‘𝐶) supp ∅) = {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
17	16	releqd 5685	. . . 4 ⊢ (𝐶 ∈ Cat → (Rel ((Iso‘𝐶) supp ∅) ↔ Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅}))
18	8, 17	mpbird 256	. . 3 ⊢ (𝐶 ∈ Cat → Rel ((Iso‘𝐶) supp ∅))
19		cicfval 17498	. . . 4 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅))
20	19	releqd 5685	. . 3 ⊢ (𝐶 ∈ Cat → (Rel ( ≃𝑐 ‘𝐶) ↔ Rel ((Iso‘𝐶) supp ∅)))
21	18, 20	mpbird 256	. 2 ⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶))
22		cicsym 17505	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐 ‘𝐶)𝑦) → 𝑦( ≃𝑐 ‘𝐶)𝑥)
23		cictr 17506	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐 ‘𝐶)𝑦 ∧ 𝑦( ≃𝑐 ‘𝐶)𝑧) → 𝑥( ≃𝑐 ‘𝐶)𝑧)
24	23	3expb 1119	. 2 ⊢ ((𝐶 ∈ Cat ∧ (𝑥( ≃𝑐 ‘𝐶)𝑦 ∧ 𝑦( ≃𝑐 ‘𝐶)𝑧)) → 𝑥( ≃𝑐 ‘𝐶)𝑧)
25		cicref 17502	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥( ≃𝑐 ‘𝐶)𝑥)
26		ciclcl 17503	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐 ‘𝐶)𝑥) → 𝑥 ∈ (Base‘𝐶))
27	25, 26	impbida 798	. 2 ⊢ (𝐶 ∈ Cat → (𝑥 ∈ (Base‘𝐶) ↔ 𝑥( ≃𝑐 ‘𝐶)𝑥))
28	21, 22, 24, 27	iserd 8513	1 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) Er (Base‘𝐶))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  {crab 3068  Vcvv 3431  ∅c0 4258  ⟨cop 4569   class class class wbr 5075  {copab 5137   × cxp 5584  Rel wrel 5591   Fn wfn 6423  ‘cfv 6428  (class class class)co 7269   supp csupp 7966   Er wer 8484  Basecbs 16901  Catccat 17362  Isociso 17447   ≃_𝑐 ccic 17496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5210  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7580

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5486  df-xp 5592  df-rel 5593  df-cnv 5594  df-co 5595  df-dm 5596  df-rn 5597  df-res 5598  df-ima 5599  df-iota 6386  df-fun 6430  df-fn 6431  df-f 6432  df-f1 6433  df-fo 6434  df-f1o 6435  df-fv 6436  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-1st 7822  df-2nd 7823  df-supp 7967  df-er 8487  df-cat 17366  df-cid 17367  df-sect 17448  df-inv 17449  df-iso 17450  df-cic 17497

This theorem is referenced by: (None)