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Theorem cicer 17518
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 5731 . . . . . 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 6774 . . . . . . . . 9 (𝑓 = ⟨𝑥, 𝑦⟩ → ((Iso‘𝐶)‘𝑓) = ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩))
43neeq1d 3003 . . . . . . . 8 (𝑓 = ⟨𝑥, 𝑦⟩ → (((Iso‘𝐶)‘𝑓) ≠ ∅ ↔ ((Iso‘𝐶)‘⟨𝑥, 𝑦⟩) ≠ ∅))
54rabxp 5635 . . . . . . 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 5689 . . . . 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 256 . . . 4 (𝐶 ∈ Cat → Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})
9 isofn 17487 . . . . . 6 (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
10 fvex 6787 . . . . . . 7 (Base‘𝐶) ∈ V
11 sqxpexg 7605 . . . . . . 7 ((Base‘𝐶) ∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
1210, 11mp1i 13 . . . . . 6 (𝐶 ∈ Cat → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
13 0ex 5231 . . . . . . 7 ∅ ∈ V
1413a1i 11 . . . . . 6 (𝐶 ∈ Cat → ∅ ∈ V)
15 suppvalfn 7985 . . . . . 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 5689 . . . 4 (𝐶 ∈ Cat → (Rel ((Iso‘𝐶) supp ∅) ↔ Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅}))
188, 17mpbird 256 . . 3 (𝐶 ∈ Cat → Rel ((Iso‘𝐶) supp ∅))
19 cicfval 17509 . . . 4 (𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
2019releqd 5689 . . 3 (𝐶 ∈ Cat → (Rel ( ≃𝑐𝐶) ↔ Rel ((Iso‘𝐶) supp ∅)))
2118, 20mpbird 256 . 2 (𝐶 ∈ Cat → Rel ( ≃𝑐𝐶))
22 cicsym 17516 . 2 ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐𝐶)𝑦) → 𝑦( ≃𝑐𝐶)𝑥)
23 cictr 17517 . . 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 17513 . . 3 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥( ≃𝑐𝐶)𝑥)
26 ciclcl 17514 . . 3 ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐𝐶)𝑥) → 𝑥 ∈ (Base‘𝐶))
2725, 26impbida 798 . 2 (𝐶 ∈ Cat → (𝑥 ∈ (Base‘𝐶) ↔ 𝑥( ≃𝑐𝐶)𝑥))
2821, 22, 24, 27iserd 8524 1 (𝐶 ∈ Cat → ( ≃𝑐𝐶) Er (Base‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1539  wcel 2106  wne 2943  {crab 3068  Vcvv 3432  c0 4256  cop 4567   class class class wbr 5074  {copab 5136   × cxp 5587  Rel wrel 5594   Fn wfn 6428  cfv 6433  (class class class)co 7275   supp csupp 7977   Er wer 8495  Basecbs 16912  Catccat 17373  Isociso 17458  𝑐 ccic 17507
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 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
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 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-supp 7978  df-er 8498  df-cat 17377  df-cid 17378  df-sect 17459  df-inv 17460  df-iso 17461  df-cic 17508
This theorem is referenced by: (None)
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