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Theorem cicer 17854
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 relopabv 5834 . . . . . 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 6907 . . . . . . . . 9 (𝑓 = ⟨𝑥, 𝑦⟩ → ((Iso‘𝐶)‘𝑓) = ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩))
43neeq1d 2998 . . . . . . . 8 (𝑓 = ⟨𝑥, 𝑦⟩ → (((Iso‘𝐶)‘𝑓) ≠ ∅ ↔ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅))
54rabxp 5737 . . . . . . 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 5791 . . . . 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 257 . . . 4 (𝐶 ∈ Cat → Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
9 isofn 17823 . . . . . 6 (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
10 fvex 6920 . . . . . . 7 (Base‘𝐶) ∈ V
11 sqxpexg 7774 . . . . . . 7 ((Base‘𝐶) ∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
1210, 11mp1i 13 . . . . . 6 (𝐶 ∈ Cat → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
13 0ex 5313 . . . . . . 7 ∅ ∈ V
1413a1i 11 . . . . . 6 (𝐶 ∈ Cat → ∅ ∈ V)
15 suppvalfn 8192 . . . . . 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 1370 . . . . 5 (𝐶 ∈ Cat → ((Iso‘𝐶) supp ∅) = {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
1716releqd 5791 . . . 4 (𝐶 ∈ Cat → (Rel ((Iso‘𝐶) supp ∅) ↔ Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅}))
188, 17mpbird 257 . . 3 (𝐶 ∈ Cat → Rel ((Iso‘𝐶) supp ∅))
19 cicfval 17845 . . . 4 (𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
2019releqd 5791 . . 3 (𝐶 ∈ Cat → (Rel ( ≃𝑐𝐶) ↔ Rel ((Iso‘𝐶) supp ∅)))
2118, 20mpbird 257 . 2 (𝐶 ∈ Cat → Rel ( ≃𝑐𝐶))
22 cicsym 17852 . 2 ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐𝐶)𝑦) → 𝑦( ≃𝑐𝐶)𝑥)
23 cictr 17853 . . 3 ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐𝐶)𝑦𝑦( ≃𝑐𝐶)𝑧) → 𝑥( ≃𝑐𝐶)𝑧)
24233expb 1119 . 2 ((𝐶 ∈ Cat ∧ (𝑥( ≃𝑐𝐶)𝑦𝑦( ≃𝑐𝐶)𝑧)) → 𝑥( ≃𝑐𝐶)𝑧)
25 cicref 17849 . . 3 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥( ≃𝑐𝐶)𝑥)
26 ciclcl 17850 . . 3 ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐𝐶)𝑥) → 𝑥 ∈ (Base‘𝐶))
2725, 26impbida 801 . 2 (𝐶 ∈ Cat → (𝑥 ∈ (Base‘𝐶) ↔ 𝑥( ≃𝑐𝐶)𝑥))
2821, 22, 24, 27iserd 8770 1 (𝐶 ∈ Cat → ( ≃𝑐𝐶) Er (Base‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  wne 2938  {crab 3433  Vcvv 3478  c0 4339  cop 4637   class class class wbr 5148  {copab 5210   × cxp 5687  Rel wrel 5694   Fn wfn 6558  cfv 6563  (class class class)co 7431   supp csupp 8184   Er wer 8741  Basecbs 17245  Catccat 17709  Isociso 17794  𝑐 ccic 17843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-supp 8185  df-er 8744  df-cat 17713  df-cid 17714  df-sect 17795  df-inv 17796  df-iso 17797  df-cic 17844
This theorem is referenced by: (None)
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