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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 4337 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅) | |
2 | homarcl.h | . . . . 5 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | df-homa 18022 | . . . . . 6 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) | |
4 | 3 | fvmptndm 7041 | . . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅) |
5 | 2, 4 | eqtrid 2780 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅) |
6 | 5 | oveqd 7443 | . . 3 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = (𝑋∅𝑌)) |
7 | 0ov 7463 | . . 3 ⊢ (𝑋∅𝑌) = ∅ | |
8 | 6, 7 | eqtrdi 2784 | . 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 1533 ∈ wcel 2098 ∅c0 4326 {csn 4632 ↦ cmpt 5235 × cxp 5680 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 Hom chom 17251 Catccat 17651 Homachoma 18019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-dm 5692 df-iota 6505 df-fv 6561 df-ov 7429 df-homa 18022 |
This theorem is referenced by: homarcl2 18031 homarel 18032 homa1 18033 homahom2 18034 coahom 18066 arwlid 18068 arwrid 18069 arwass 18070 |
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