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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 4264 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅) | |
2 | homarcl.h | . . . . 5 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | df-homa 17562 | . . . . . 6 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) | |
4 | 3 | fvmptndm 6869 | . . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅) |
5 | 2, 4 | eqtrid 2791 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅) |
6 | 5 | oveqd 7251 | . . 3 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = (𝑋∅𝑌)) |
7 | 0ov 7271 | . . 3 ⊢ (𝑋∅𝑌) = ∅ | |
8 | 6, 7 | eqtrdi 2796 | . 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 1543 ∈ wcel 2112 ∅c0 4253 {csn 4557 ↦ cmpt 5151 × cxp 5566 ‘cfv 6400 (class class class)co 7234 Basecbs 16790 Hom chom 16843 Catccat 17197 Homachoma 17559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pr 5338 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3885 df-un 3887 df-in 3889 df-ss 3899 df-nul 4254 df-if 4456 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4836 df-br 5070 df-opab 5132 df-mpt 5152 df-dm 5578 df-iota 6358 df-fv 6408 df-ov 7237 df-homa 17562 |
This theorem is referenced by: homarcl2 17571 homarel 17572 homa1 17573 homahom2 17574 coahom 17606 arwlid 17608 arwrid 17609 arwass 17610 |
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