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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 17740 | . . . 4 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) |
4 | 3 | breqd 5158 | . 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ 𝑋((Iso‘𝐶) supp ∅)𝑌)) |
5 | df-br 5148 | . . 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 6890 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝑌〉) = ((Iso‘𝐶)‘〈𝑋, 𝑌〉)) |
10 | 9 | neeq1d 3001 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝑌〉) ≠ ∅ ↔ ((Iso‘𝐶)‘〈𝑋, 𝑌〉) ≠ ∅)) |
11 | df-ov 7407 | . . . . . 6 ⊢ (𝑋𝐼𝑌) = (𝐼‘〈𝑋, 𝑌〉) | |
12 | 11 | eqcomi 2742 | . . . . 5 ⊢ (𝐼‘〈𝑋, 𝑌〉) = (𝑋𝐼𝑌) |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝑌〉) = (𝑋𝐼𝑌)) |
14 | 13 | neeq1d 3001 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝑌〉) ≠ ∅ ↔ (𝑋𝐼𝑌) ≠ ∅)) |
15 | fvexd 6903 | . . . . 5 ⊢ (𝜑 → (Base‘𝐶) ∈ V) | |
16 | 15, 15 | xpexd 7733 | . . . 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 2844 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
20 | cic.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
21 | 20, 18 | eleqtrdi 2844 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
22 | 19, 21 | opelxpd 5713 | . . . 4 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ ((Base‘𝐶) × (Base‘𝐶))) |
23 | isofn 17718 | . . . . 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 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 ∅))) | |
26 | 16, 22, 24, 25 | syl3anc 1372 | . . 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 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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