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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 4333 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅) | |
2 | homarcl.h | . . . . 5 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | df-homa 18018 | . . . . . 6 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) | |
4 | 3 | fvmptndm 7035 | . . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅) |
5 | 2, 4 | eqtrid 2777 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅) |
6 | 5 | oveqd 7436 | . . 3 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = (𝑋∅𝑌)) |
7 | 0ov 7456 | . . 3 ⊢ (𝑋∅𝑌) = ∅ | |
8 | 6, 7 | eqtrdi 2781 | . 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 4322 {csn 4630 ↦ cmpt 5232 × cxp 5676 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 Hom chom 17247 Catccat 17647 Homachoma 18015 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-dm 5688 df-iota 6501 df-fv 6557 df-ov 7422 df-homa 18018 |
This theorem is referenced by: homarcl2 18027 homarel 18028 homa1 18029 homahom2 18030 coahom 18062 arwlid 18064 arwrid 18065 arwass 18066 |
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