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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 4340 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅) | |
| 2 | homarcl.h | . . . . 5 ⊢ 𝐻 = (Homa‘𝐶) | |
| 3 | df-homa 18071 | . . . . . 6 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) | |
| 4 | 3 | fvmptndm 7047 | . . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅) |
| 5 | 2, 4 | eqtrid 2789 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅) |
| 6 | 5 | oveqd 7448 | . . 3 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = (𝑋∅𝑌)) |
| 7 | 0ov 7468 | . . 3 ⊢ (𝑋∅𝑌) = ∅ | |
| 8 | 6, 7 | eqtrdi 2793 | . 2 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = ∅) |
| 9 | 1, 8 | nsyl2 141 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ∅c0 4333 {csn 4626 ↦ cmpt 5225 × cxp 5683 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Hom chom 17308 Catccat 17707 Homachoma 18068 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434 df-homa 18071 |
| This theorem is referenced by: homarcl2 18080 homarel 18081 homa1 18082 homahom2 18083 coahom 18115 arwlid 18117 arwrid 18118 arwass 18119 |
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