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Mirrors > Home > MPE Home > Th. List > homarcl2 | Structured version Visualization version GIF version |
Description: Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homahom.h | ⊢ 𝐻 = (Homa‘𝐶) |
homarcl2.b | ⊢ 𝐵 = (Base‘𝐶) |
Ref | Expression |
---|---|
homarcl2 | ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6382 | . . . 4 ⊢ (𝐹 ∈ (𝐻‘〈𝑋, 𝑌〉) → 〈𝑋, 𝑌〉 ∈ dom 𝐻) | |
2 | df-ov 6817 | . . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
3 | 1, 2 | eleq2s 2857 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 〈𝑋, 𝑌〉 ∈ dom 𝐻) |
4 | homahom.h | . . . . 5 ⊢ 𝐻 = (Homa‘𝐶) | |
5 | homarcl2.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
6 | 4 | homarcl 16899 | . . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
7 | 4, 5, 6 | homaf 16901 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V)) |
8 | fdm 6212 | . . . 4 ⊢ (𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V) → dom 𝐻 = (𝐵 × 𝐵)) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → dom 𝐻 = (𝐵 × 𝐵)) |
10 | 3, 9 | eleqtrd 2841 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
11 | opelxp 5303 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵) ↔ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) | |
12 | 10, 11 | sylib 208 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 𝒫 cpw 4302 〈cop 4327 × cxp 5264 dom cdm 5266 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 Basecbs 16079 Homachoma 16894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6817 df-homa 16897 |
This theorem is referenced by: homarel 16907 homa1 16908 homahom2 16909 homadm 16911 homacd 16912 arwdm 16918 arwcd 16919 coahom 16941 arwlid 16943 arwrid 16944 arwass 16945 |
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