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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 16457 | . . . 4 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) |
4 | 3 | breqd 4664 | . 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ 𝑋((Iso‘𝐶) supp ∅)𝑌)) |
5 | df-br 4654 | . . 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 6193 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝑌〉) = ((Iso‘𝐶)‘〈𝑋, 𝑌〉)) |
10 | 9 | neeq1d 2853 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝑌〉) ≠ ∅ ↔ ((Iso‘𝐶)‘〈𝑋, 𝑌〉) ≠ ∅)) |
11 | df-ov 6653 | . . . . . 6 ⊢ (𝑋𝐼𝑌) = (𝐼‘〈𝑋, 𝑌〉) | |
12 | 11 | eqcomi 2631 | . . . . 5 ⊢ (𝐼‘〈𝑋, 𝑌〉) = (𝑋𝐼𝑌) |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝑌〉) = (𝑋𝐼𝑌)) |
14 | 13 | neeq1d 2853 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝑌〉) ≠ ∅ ↔ (𝑋𝐼𝑌) ≠ ∅)) |
15 | fvexd 6203 | . . . . 5 ⊢ (𝜑 → (Base‘𝐶) ∈ V) | |
16 | sqxpexg 6963 | . . . . 5 ⊢ ((Base‘𝐶) ∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ V) |
18 | cic.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
19 | cic.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
20 | 18, 19 | syl6eleq 2711 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
21 | cic.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
22 | 21, 19 | syl6eleq 2711 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
23 | opelxp 5146 | . . . . 5 ⊢ (〈𝑋, 𝑌〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↔ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) | |
24 | 20, 22, 23 | sylanbrc 698 | . . . 4 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ ((Base‘𝐶) × (Base‘𝐶))) |
25 | isofn 16435 | . . . . 5 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | |
26 | 1, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) |
27 | fvn0elsuppb 7312 | . . . 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 ∅))) | |
28 | 17, 24, 26, 27 | syl3anc 1326 | . . 3 ⊢ (𝜑 → (((Iso‘𝐶)‘〈𝑋, 𝑌〉) ≠ ∅ ↔ 〈𝑋, 𝑌〉 ∈ ((Iso‘𝐶) supp ∅))) |
29 | 10, 14, 28 | 3bitr3rd 299 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ ((Iso‘𝐶) supp ∅) ↔ (𝑋𝐼𝑌) ≠ ∅)) |
30 | 4, 6, 29 | 3bitrd 294 | 1 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ (𝑋𝐼𝑌) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∅c0 3915 〈cop 4183 class class class wbr 4653 × cxp 5112 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 supp csupp 7295 Basecbs 15857 Catccat 16325 Isociso 16406 ≃𝑐 ccic 16455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-supp 7296 df-inv 16408 df-iso 16409 df-cic 16456 |
This theorem is referenced by: cic 16459 |
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