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Theorem brcic 17303
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 17302 . . . 4 (𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
31, 2syl 17 . . 3 (𝜑 → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
43breqd 5064 . 2 (𝜑 → (𝑋( ≃𝑐𝐶)𝑌𝑋((Iso‘𝐶) supp ∅)𝑌))
5 df-br 5054 . . 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 6719 . . . 4 (𝜑 → (𝐼‘⟨𝑋, 𝑌⟩) = ((Iso‘𝐶)‘⟨𝑋, 𝑌⟩))
109neeq1d 3000 . . 3 (𝜑 → ((𝐼‘⟨𝑋, 𝑌⟩) ≠ ∅ ↔ ((Iso‘𝐶)‘⟨𝑋, 𝑌⟩) ≠ ∅))
11 df-ov 7216 . . . . . 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 6732 . . . . 5 (𝜑 → (Base‘𝐶) ∈ V)
1615, 15xpexd 7536 . . . 4 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
17 cic.x . . . . . 6 (𝜑𝑋𝐵)
18 cic.b . . . . . 6 𝐵 = (Base‘𝐶)
1917, 18eleqtrdi 2848 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
20 cic.y . . . . . 6 (𝜑𝑌𝐵)
2120, 18eleqtrdi 2848 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
2219, 21opelxpd 5589 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((Base‘𝐶) × (Base‘𝐶)))
23 isofn 17280 . . . . 5 (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
241, 23syl 17 . . . 4 (𝜑 → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
25 fvn0elsuppb 7923 . . . 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 313 . 2 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ ((Iso‘𝐶) supp ∅) ↔ (𝑋𝐼𝑌) ≠ ∅))
284, 6, 273bitrd 308 1 (𝜑 → (𝑋( ≃𝑐𝐶)𝑌 ↔ (𝑋𝐼𝑌) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wcel 2110  wne 2940  Vcvv 3408  c0 4237  cop 4547   class class class wbr 5053   × cxp 5549   Fn wfn 6375  cfv 6380  (class class class)co 7213   supp csupp 7903  Basecbs 16760  Catccat 17167  Isociso 17251  𝑐 ccic 17300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-supp 7904  df-inv 17253  df-iso 17254  df-cic 17301
This theorem is referenced by:  cic  17304
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