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Theorem brcic 17741
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 17740 . . . 4 (𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
31, 2syl 17 . . 3 (𝜑 → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
43breqd 5158 . 2 (𝜑 → (𝑋( ≃𝑐𝐶)𝑌𝑋((Iso‘𝐶) supp ∅)𝑌))
5 df-br 5148 . . 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 6890 . . . 4 (𝜑 → (𝐼‘⟨𝑋, 𝑌⟩) = ((Iso‘𝐶)‘⟨𝑋, 𝑌⟩))
109neeq1d 3001 . . 3 (𝜑 → ((𝐼‘⟨𝑋, 𝑌⟩) ≠ ∅ ↔ ((Iso‘𝐶)‘⟨𝑋, 𝑌⟩) ≠ ∅))
11 df-ov 7407 . . . . . 6 (𝑋𝐼𝑌) = (𝐼‘⟨𝑋, 𝑌⟩)
1211eqcomi 2742 . . . . 5 (𝐼‘⟨𝑋, 𝑌⟩) = (𝑋𝐼𝑌)
1312a1i 11 . . . 4 (𝜑 → (𝐼‘⟨𝑋, 𝑌⟩) = (𝑋𝐼𝑌))
1413neeq1d 3001 . . 3 (𝜑 → ((𝐼‘⟨𝑋, 𝑌⟩) ≠ ∅ ↔ (𝑋𝐼𝑌) ≠ ∅))
15 fvexd 6903 . . . . 5 (𝜑 → (Base‘𝐶) ∈ V)
1615, 15xpexd 7733 . . . 4 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
17 cic.x . . . . . 6 (𝜑𝑋𝐵)
18 cic.b . . . . . 6 𝐵 = (Base‘𝐶)
1917, 18eleqtrdi 2844 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
20 cic.y . . . . . 6 (𝜑𝑌𝐵)
2120, 18eleqtrdi 2844 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
2219, 21opelxpd 5713 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((Base‘𝐶) × (Base‘𝐶)))
23 isofn 17718 . . . . 5 (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
241, 23syl 17 . . . 4 (𝜑 → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
25 fvn0elsuppb 8161 . . . 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 1372 . . 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 205   = wceq 1542  wcel 2107  wne 2941  Vcvv 3475  c0 4321  cop 4633   class class class wbr 5147   × cxp 5673   Fn wfn 6535  cfv 6540  (class class class)co 7404   supp csupp 8141  Basecbs 17140  Catccat 17604  Isociso 17689  𝑐 ccic 17738
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7970  df-2nd 7971  df-supp 8142  df-inv 17691  df-iso 17692  df-cic 17739
This theorem is referenced by:  cic  17742
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