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Theorem brcic 17842
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 17841 . . . 4 (𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
31, 2syl 17 . . 3 (𝜑 → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
43breqd 5154 . 2 (𝜑 → (𝑋( ≃𝑐𝐶)𝑌𝑋((Iso‘𝐶) supp ∅)𝑌))
5 df-br 5144 . . 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 6908 . . . 4 (𝜑 → (𝐼‘⟨𝑋, 𝑌⟩) = ((Iso‘𝐶)‘⟨𝑋, 𝑌⟩))
109neeq1d 3000 . . 3 (𝜑 → ((𝐼‘⟨𝑋, 𝑌⟩) ≠ ∅ ↔ ((Iso‘𝐶)‘⟨𝑋, 𝑌⟩) ≠ ∅))
11 df-ov 7434 . . . . . 6 (𝑋𝐼𝑌) = (𝐼‘⟨𝑋, 𝑌⟩)
1211eqcomi 2746 . . . . 5 (𝐼‘⟨𝑋, 𝑌⟩) = (𝑋𝐼𝑌)
1312a1i 11 . . . 4 (𝜑 → (𝐼‘⟨𝑋, 𝑌⟩) = (𝑋𝐼𝑌))
1413neeq1d 3000 . . 3 (𝜑 → ((𝐼‘⟨𝑋, 𝑌⟩) ≠ ∅ ↔ (𝑋𝐼𝑌) ≠ ∅))
15 fvexd 6921 . . . . 5 (𝜑 → (Base‘𝐶) ∈ V)
1615, 15xpexd 7771 . . . 4 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
17 cic.x . . . . . 6 (𝜑𝑋𝐵)
18 cic.b . . . . . 6 𝐵 = (Base‘𝐶)
1917, 18eleqtrdi 2851 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
20 cic.y . . . . . 6 (𝜑𝑌𝐵)
2120, 18eleqtrdi 2851 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
2219, 21opelxpd 5724 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((Base‘𝐶) × (Base‘𝐶)))
23 isofn 17819 . . . . 5 (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
241, 23syl 17 . . . 4 (𝜑 → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
25 fvn0elsuppb 8206 . . . 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 1373 . . 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 1540  wcel 2108  wne 2940  Vcvv 3480  c0 4333  cop 4632   class class class wbr 5143   × cxp 5683   Fn wfn 6556  cfv 6561  (class class class)co 7431   supp csupp 8185  Basecbs 17247  Catccat 17707  Isociso 17790  𝑐 ccic 17839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-supp 8186  df-inv 17792  df-iso 17793  df-cic 17840
This theorem is referenced by:  cic  17843
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