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Theorem cicer 16687
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 → ( ≃𝑐𝐶) Er (Base‘𝐶))

Proof of Theorem cicer
Dummy variables 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 5403 . . . . . 6 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)}
21a1i 11 . . . . 5 (𝐶 ∈ Cat → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)})
3 fveq2 6353 . . . . . . . . 9 (𝑓 = ⟨𝑥, 𝑦⟩ → ((Iso‘𝐶)‘𝑓) = ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩))
43neeq1d 2991 . . . . . . . 8 (𝑓 = ⟨𝑥, 𝑦⟩ → (((Iso‘𝐶)‘𝑓) ≠ ∅ ↔ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅))
54rabxp 5311 . . . . . . 7 {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)}
65a1i 11 . . . . . 6 (𝐶 ∈ Cat → {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅)})
76releqd 5360 . . . . 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⟩) ≠ ∅)}))
82, 7mpbird 247 . . . 4 (𝐶 ∈ Cat → Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
9 isofn 16656 . . . . . 6 (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
10 fvex 6363 . . . . . . 7 (Base‘𝐶) ∈ V
11 sqxpexg 7129 . . . . . . 7 ((Base‘𝐶) ∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
1210, 11mp1i 13 . . . . . 6 (𝐶 ∈ Cat → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
13 0ex 4942 . . . . . . 7 ∅ ∈ V
1413a1i 11 . . . . . 6 (𝐶 ∈ Cat → ∅ ∈ V)
15 suppvalfn 7471 . . . . . 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) ≠ ∅})
169, 12, 14, 15syl3anc 1477 . . . . 5 (𝐶 ∈ Cat → ((Iso‘𝐶) supp ∅) = {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
1716releqd 5360 . . . 4 (𝐶 ∈ Cat → (Rel ((Iso‘𝐶) supp ∅) ↔ Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅}))
188, 17mpbird 247 . . 3 (𝐶 ∈ Cat → Rel ((Iso‘𝐶) supp ∅))
19 cicfval 16678 . . . 4 (𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
2019releqd 5360 . . 3 (𝐶 ∈ Cat → (Rel ( ≃𝑐𝐶) ↔ Rel ((Iso‘𝐶) supp ∅)))
2118, 20mpbird 247 . 2 (𝐶 ∈ Cat → Rel ( ≃𝑐𝐶))
22 cicsym 16685 . 2 ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐𝐶)𝑦) → 𝑦( ≃𝑐𝐶)𝑥)
23 cictr 16686 . . 3 ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐𝐶)𝑦𝑦( ≃𝑐𝐶)𝑧) → 𝑥( ≃𝑐𝐶)𝑧)
24233expb 1114 . 2 ((𝐶 ∈ Cat ∧ (𝑥( ≃𝑐𝐶)𝑦𝑦( ≃𝑐𝐶)𝑧)) → 𝑥( ≃𝑐𝐶)𝑧)
25 cicref 16682 . . 3 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥( ≃𝑐𝐶)𝑥)
26 ciclcl 16683 . . 3 ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐𝐶)𝑥) → 𝑥 ∈ (Base‘𝐶))
2725, 26impbida 913 . 2 (𝐶 ∈ Cat → (𝑥 ∈ (Base‘𝐶) ↔ 𝑥( ≃𝑐𝐶)𝑥))
2821, 22, 24, 27iserd 7939 1 (𝐶 ∈ Cat → ( ≃𝑐𝐶) Er (Base‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  wcel 2139  wne 2932  {crab 3054  Vcvv 3340  c0 4058  cop 4327   class class class wbr 4804  {copab 4864   × cxp 5264  Rel wrel 5271   Fn wfn 6044  cfv 6049  (class class class)co 6814   supp csupp 7464   Er wer 7910  Basecbs 16079  Catccat 16546  Isociso 16627  𝑐 ccic 16676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-supp 7465  df-er 7913  df-cat 16550  df-cid 16551  df-sect 16628  df-inv 16629  df-iso 16630  df-cic 16677
This theorem is referenced by: (None)
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