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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 4296 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅) | |
2 | homarcl.h | . . . . 5 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | df-homa 17274 | . . . . . 6 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) | |
4 | 3 | fvmptndm 6790 | . . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅) |
5 | 2, 4 | syl5eq 2865 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅) |
6 | 5 | oveqd 7162 | . . 3 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = (𝑋∅𝑌)) |
7 | 0ov 7182 | . . 3 ⊢ (𝑋∅𝑌) = ∅ | |
8 | 6, 7 | syl6eq 2869 | . 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 1528 ∈ wcel 2105 ∅c0 4288 {csn 4557 ↦ cmpt 5137 × cxp 5546 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Hom chom 16564 Catccat 16923 Homachoma 17271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-dm 5558 df-iota 6307 df-fv 6356 df-ov 7148 df-homa 17274 |
This theorem is referenced by: homarcl2 17283 homarel 17284 homa1 17285 homahom2 17286 coahom 17318 arwlid 17320 arwrid 17321 arwass 17322 |
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