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Theorem brcic 17846
Description: The relation "is isomorphic to" for categories. (Contributed by AV, 5-Apr-2020.)
Hypotheses
Ref Expression
cic.i 𝐼 = (Iso‘𝐶)
cic.b 𝐵 = (Base‘𝐶)
cic.c (𝜑𝐶 ∈ Cat)
cic.x (𝜑𝑋𝐵)
cic.y (𝜑𝑌𝐵)
Assertion
Ref Expression
brcic (𝜑 → (𝑋( ≃𝑐𝐶)𝑌 ↔ (𝑋𝐼𝑌) ≠ ∅))

Proof of Theorem brcic
StepHypRef Expression
1 cic.c . . . 4 (𝜑𝐶 ∈ Cat)
2 cicfval 17845 . . . 4 (𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
31, 2syl 17 . . 3 (𝜑 → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
43breqd 5159 . 2 (𝜑 → (𝑋( ≃𝑐𝐶)𝑌𝑋((Iso‘𝐶) supp ∅)𝑌))
5 df-br 5149 . . 3 (𝑋((Iso‘𝐶) supp ∅)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ ((Iso‘𝐶) supp ∅))
65a1i 11 . 2 (𝜑 → (𝑋((Iso‘𝐶) supp ∅)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ ((Iso‘𝐶) supp ∅)))
7 cic.i . . . . . 6 𝐼 = (Iso‘𝐶)
87a1i 11 . . . . 5 (𝜑𝐼 = (Iso‘𝐶))
98fveq1d 6909 . . . 4 (𝜑 → (𝐼‘⟨𝑋, 𝑌⟩) = ((Iso‘𝐶)‘⟨𝑋, 𝑌⟩))
109neeq1d 2998 . . 3 (𝜑 → ((𝐼‘⟨𝑋, 𝑌⟩) ≠ ∅ ↔ ((Iso‘𝐶)‘⟨𝑋, 𝑌⟩) ≠ ∅))
11 df-ov 7434 . . . . . 6 (𝑋𝐼𝑌) = (𝐼‘⟨𝑋, 𝑌⟩)
1211eqcomi 2744 . . . . 5 (𝐼‘⟨𝑋, 𝑌⟩) = (𝑋𝐼𝑌)
1312a1i 11 . . . 4 (𝜑 → (𝐼‘⟨𝑋, 𝑌⟩) = (𝑋𝐼𝑌))
1413neeq1d 2998 . . 3 (𝜑 → ((𝐼‘⟨𝑋, 𝑌⟩) ≠ ∅ ↔ (𝑋𝐼𝑌) ≠ ∅))
15 fvexd 6922 . . . . 5 (𝜑 → (Base‘𝐶) ∈ V)
1615, 15xpexd 7770 . . . 4 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
17 cic.x . . . . . 6 (𝜑𝑋𝐵)
18 cic.b . . . . . 6 𝐵 = (Base‘𝐶)
1917, 18eleqtrdi 2849 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
20 cic.y . . . . . 6 (𝜑𝑌𝐵)
2120, 18eleqtrdi 2849 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
2219, 21opelxpd 5728 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((Base‘𝐶) × (Base‘𝐶)))
23 isofn 17823 . . . . 5 (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
241, 23syl 17 . . . 4 (𝜑 → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
25 fvn0elsuppb 8205 . . . 4 ((((Base‘𝐶) × (Base‘𝐶)) ∈ V ∧ ⟨𝑋, 𝑌⟩ ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) → (((Iso‘𝐶)‘⟨𝑋, 𝑌⟩) ≠ ∅ ↔ ⟨𝑋, 𝑌⟩ ∈ ((Iso‘𝐶) supp ∅)))
2616, 22, 24, 25syl3anc 1370 . . 3 (𝜑 → (((Iso‘𝐶)‘⟨𝑋, 𝑌⟩) ≠ ∅ ↔ ⟨𝑋, 𝑌⟩ ∈ ((Iso‘𝐶) supp ∅)))
2710, 14, 263bitr3rd 310 . 2 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ ((Iso‘𝐶) supp ∅) ↔ (𝑋𝐼𝑌) ≠ ∅))
284, 6, 273bitrd 305 1 (𝜑 → (𝑋( ≃𝑐𝐶)𝑌 ↔ (𝑋𝐼𝑌) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  c0 4339  cop 4637   class class class wbr 5148   × cxp 5687   Fn wfn 6558  cfv 6563  (class class class)co 7431   supp csupp 8184  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-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-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-supp 8185  df-inv 17796  df-iso 17797  df-cic 17844
This theorem is referenced by:  cic  17847
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