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Mirrors > Home > MPE Home > Th. List > brcic | Structured version Visualization version GIF version |
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 17069 | . . . 4 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) |
4 | 3 | breqd 5079 | . 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ 𝑋((Iso‘𝐶) supp ∅)𝑌)) |
5 | df-br 5069 | . . 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 6674 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝑌〉) = ((Iso‘𝐶)‘〈𝑋, 𝑌〉)) |
10 | 9 | neeq1d 3077 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝑌〉) ≠ ∅ ↔ ((Iso‘𝐶)‘〈𝑋, 𝑌〉) ≠ ∅)) |
11 | df-ov 7161 | . . . . . 6 ⊢ (𝑋𝐼𝑌) = (𝐼‘〈𝑋, 𝑌〉) | |
12 | 11 | eqcomi 2832 | . . . . 5 ⊢ (𝐼‘〈𝑋, 𝑌〉) = (𝑋𝐼𝑌) |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝑌〉) = (𝑋𝐼𝑌)) |
14 | 13 | neeq1d 3077 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝑌〉) ≠ ∅ ↔ (𝑋𝐼𝑌) ≠ ∅)) |
15 | fvexd 6687 | . . . . 5 ⊢ (𝜑 → (Base‘𝐶) ∈ V) | |
16 | 15, 15 | xpexd 7476 | . . . 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 2925 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
20 | cic.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
21 | 20, 18 | eleqtrdi 2925 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
22 | 19, 21 | opelxpd 5595 | . . . 4 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ ((Base‘𝐶) × (Base‘𝐶))) |
23 | isofn 17047 | . . . . 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 7849 | . . . 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 1367 | . . 3 ⊢ (𝜑 → (((Iso‘𝐶)‘〈𝑋, 𝑌〉) ≠ ∅ ↔ 〈𝑋, 𝑌〉 ∈ ((Iso‘𝐶) supp ∅))) |
27 | 10, 14, 26 | 3bitr3rd 312 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ ((Iso‘𝐶) supp ∅) ↔ (𝑋𝐼𝑌) ≠ ∅)) |
28 | 4, 6, 27 | 3bitrd 307 | 1 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ (𝑋𝐼𝑌) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∅c0 4293 〈cop 4575 class class class wbr 5068 × cxp 5555 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 supp csupp 7832 Basecbs 16485 Catccat 16937 Isociso 17018 ≃𝑐 ccic 17067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-supp 7833 df-inv 17020 df-iso 17021 df-cic 17068 |
This theorem is referenced by: cic 17071 |
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