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Mirrors > Home > MPE Home > Th. List > homarcl | Structured version Visualization version GIF version |
Description: Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homarcl.h | ⊢ 𝐻 = (Homa‘𝐶) |
Ref | Expression |
---|---|
homarcl | ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4299 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅) | |
2 | homarcl.h | . . . . 5 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | df-homa 17280 | . . . . . 6 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) | |
4 | 3 | fvmptndm 6793 | . . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅) |
5 | 2, 4 | syl5eq 2868 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅) |
6 | 5 | oveqd 7167 | . . 3 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = (𝑋∅𝑌)) |
7 | 0ov 7187 | . . 3 ⊢ (𝑋∅𝑌) = ∅ | |
8 | 6, 7 | syl6eq 2872 | . 2 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = ∅) |
9 | 1, 8 | nsyl2 143 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2110 ∅c0 4291 {csn 4561 ↦ cmpt 5139 × cxp 5548 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 Hom chom 16570 Catccat 16929 Homachoma 17277 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-dm 5560 df-iota 6309 df-fv 6358 df-ov 7153 df-homa 17280 |
This theorem is referenced by: homarcl2 17289 homarel 17290 homa1 17291 homahom2 17292 coahom 17324 arwlid 17326 arwrid 17327 arwass 17328 |
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