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Theorem brcic 16679
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 16678 . . . 4 (𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
31, 2syl 17 . . 3 (𝜑 → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
43breqd 4815 . 2 (𝜑 → (𝑋( ≃𝑐𝐶)𝑌𝑋((Iso‘𝐶) supp ∅)𝑌))
5 df-br 4805 . . 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 6355 . . . 4 (𝜑 → (𝐼‘⟨𝑋, 𝑌⟩) = ((Iso‘𝐶)‘⟨𝑋, 𝑌⟩))
109neeq1d 2991 . . 3 (𝜑 → ((𝐼‘⟨𝑋, 𝑌⟩) ≠ ∅ ↔ ((Iso‘𝐶)‘⟨𝑋, 𝑌⟩) ≠ ∅))
11 df-ov 6817 . . . . . 6 (𝑋𝐼𝑌) = (𝐼‘⟨𝑋, 𝑌⟩)
1211eqcomi 2769 . . . . 5 (𝐼‘⟨𝑋, 𝑌⟩) = (𝑋𝐼𝑌)
1312a1i 11 . . . 4 (𝜑 → (𝐼‘⟨𝑋, 𝑌⟩) = (𝑋𝐼𝑌))
1413neeq1d 2991 . . 3 (𝜑 → ((𝐼‘⟨𝑋, 𝑌⟩) ≠ ∅ ↔ (𝑋𝐼𝑌) ≠ ∅))
15 fvexd 6365 . . . . 5 (𝜑 → (Base‘𝐶) ∈ V)
16 sqxpexg 7129 . . . . 5 ((Base‘𝐶) ∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
1715, 16syl 17 . . . 4 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
18 cic.x . . . . . 6 (𝜑𝑋𝐵)
19 cic.b . . . . . 6 𝐵 = (Base‘𝐶)
2018, 19syl6eleq 2849 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
21 cic.y . . . . . 6 (𝜑𝑌𝐵)
2221, 19syl6eleq 2849 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
23 opelxp 5303 . . . . 5 (⟨𝑋, 𝑌⟩ ∈ ((Base‘𝐶) × (Base‘𝐶)) ↔ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
2420, 22, 23sylanbrc 701 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((Base‘𝐶) × (Base‘𝐶)))
25 isofn 16656 . . . . 5 (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
261, 25syl 17 . . . 4 (𝜑 → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
27 fvn0elsuppb 7481 . . . 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 ∅)))
2817, 24, 26, 27syl3anc 1477 . . 3 (𝜑 → (((Iso‘𝐶)‘⟨𝑋, 𝑌⟩) ≠ ∅ ↔ ⟨𝑋, 𝑌⟩ ∈ ((Iso‘𝐶) supp ∅)))
2910, 14, 283bitr3rd 299 . 2 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ ((Iso‘𝐶) supp ∅) ↔ (𝑋𝐼𝑌) ≠ ∅))
304, 6, 293bitrd 294 1 (𝜑 → (𝑋( ≃𝑐𝐶)𝑌 ↔ (𝑋𝐼𝑌) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2139  wne 2932  Vcvv 3340  c0 4058  cop 4327   class class class wbr 4804   × cxp 5264   Fn wfn 6044  cfv 6049  (class class class)co 6814   supp csupp 7464  Basecbs 16079  Catccat 16546  Isociso 16627  𝑐 ccic 16676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-supp 7465  df-inv 16629  df-iso 16630  df-cic 16677
This theorem is referenced by:  cic  16680
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