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| Description: The relation "is isomorphic to" for categories. (Contributed by AV, 5-Apr-2020.) | 
| Ref | Expression | 
|---|---|
| cic.i | ⊢ 𝐼 = (Iso‘𝐶) | 
| cic.b | ⊢ 𝐵 = (Base‘𝐶) | 
| cic.c | ⊢ (𝜑 → 𝐶 ∈ Cat) | 
| cic.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| cic.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) | 
| Ref | Expression | 
|---|---|
| brcic | ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ (𝑋𝐼𝑌) ≠ ∅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cic.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
| 2 | cicfval 17841 | . . . 4 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) | |
| 3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) | 
| 4 | 3 | breqd 5154 | . 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ 𝑋((Iso‘𝐶) supp ∅)𝑌)) | 
| 5 | df-br 5144 | . . 3 ⊢ (𝑋((Iso‘𝐶) supp ∅)𝑌 ↔ 〈𝑋, 𝑌〉 ∈ ((Iso‘𝐶) supp ∅)) | |
| 6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → (𝑋((Iso‘𝐶) supp ∅)𝑌 ↔ 〈𝑋, 𝑌〉 ∈ ((Iso‘𝐶) supp ∅))) | 
| 7 | cic.i | . . . . . 6 ⊢ 𝐼 = (Iso‘𝐶) | |
| 8 | 7 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐼 = (Iso‘𝐶)) | 
| 9 | 8 | fveq1d 6908 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝑌〉) = ((Iso‘𝐶)‘〈𝑋, 𝑌〉)) | 
| 10 | 9 | neeq1d 3000 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝑌〉) ≠ ∅ ↔ ((Iso‘𝐶)‘〈𝑋, 𝑌〉) ≠ ∅)) | 
| 11 | df-ov 7434 | . . . . . 6 ⊢ (𝑋𝐼𝑌) = (𝐼‘〈𝑋, 𝑌〉) | |
| 12 | 11 | eqcomi 2746 | . . . . 5 ⊢ (𝐼‘〈𝑋, 𝑌〉) = (𝑋𝐼𝑌) | 
| 13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝑌〉) = (𝑋𝐼𝑌)) | 
| 14 | 13 | neeq1d 3000 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝑌〉) ≠ ∅ ↔ (𝑋𝐼𝑌) ≠ ∅)) | 
| 15 | fvexd 6921 | . . . . 5 ⊢ (𝜑 → (Base‘𝐶) ∈ V) | |
| 16 | 15, 15 | xpexd 7771 | . . . 4 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ V) | 
| 17 | cic.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 18 | cic.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
| 19 | 17, 18 | eleqtrdi 2851 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) | 
| 20 | cic.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 21 | 20, 18 | eleqtrdi 2851 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) | 
| 22 | 19, 21 | opelxpd 5724 | . . . 4 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ ((Base‘𝐶) × (Base‘𝐶))) | 
| 23 | isofn 17819 | . . . . 5 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | |
| 24 | 1, 23 | syl 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 ∅))) | |
| 26 | 16, 22, 24, 25 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (((Iso‘𝐶)‘〈𝑋, 𝑌〉) ≠ ∅ ↔ 〈𝑋, 𝑌〉 ∈ ((Iso‘𝐶) supp ∅))) | 
| 27 | 10, 14, 26 | 3bitr3rd 310 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ ((Iso‘𝐶) supp ∅) ↔ (𝑋𝐼𝑌) ≠ ∅)) | 
| 28 | 4, 6, 27 | 3bitrd 305 | 1 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)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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