![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 17845 | . . . 4 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) |
4 | 3 | breqd 5159 | . 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ 𝑋((Iso‘𝐶) supp ∅)𝑌)) |
5 | df-br 5149 | . . 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 6909 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝑌〉) = ((Iso‘𝐶)‘〈𝑋, 𝑌〉)) |
10 | 9 | neeq1d 2998 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝑌〉) ≠ ∅ ↔ ((Iso‘𝐶)‘〈𝑋, 𝑌〉) ≠ ∅)) |
11 | df-ov 7434 | . . . . . 6 ⊢ (𝑋𝐼𝑌) = (𝐼‘〈𝑋, 𝑌〉) | |
12 | 11 | eqcomi 2744 | . . . . 5 ⊢ (𝐼‘〈𝑋, 𝑌〉) = (𝑋𝐼𝑌) |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝑌〉) = (𝑋𝐼𝑌)) |
14 | 13 | neeq1d 2998 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝑌〉) ≠ ∅ ↔ (𝑋𝐼𝑌) ≠ ∅)) |
15 | fvexd 6922 | . . . . 5 ⊢ (𝜑 → (Base‘𝐶) ∈ V) | |
16 | 15, 15 | xpexd 7770 | . . . 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 2849 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
20 | cic.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
21 | 20, 18 | eleqtrdi 2849 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
22 | 19, 21 | opelxpd 5728 | . . . 4 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ ((Base‘𝐶) × (Base‘𝐶))) |
23 | isofn 17823 | . . . . 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 8205 | . . . 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 1370 | . . 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 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∅c0 4339 〈cop 4637 class class class wbr 5148 × cxp 5687 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 supp csupp 8184 Basecbs 17245 Catccat 17709 Isociso 17794 ≃𝑐 ccic 17843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-supp 8185 df-inv 17796 df-iso 17797 df-cic 17844 |
This theorem is referenced by: cic 17847 |
Copyright terms: Public domain | W3C validator |