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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 6943 | . . . 4 ⊢ (𝐹 ∈ (𝐻‘〈𝑋, 𝑌〉) → 〈𝑋, 𝑌〉 ∈ dom 𝐻) | |
2 | df-ov 7433 | . . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
3 | 1, 2 | eleq2s 2856 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 〈𝑋, 𝑌〉 ∈ dom 𝐻) |
4 | homahom.h | . . . . 5 ⊢ 𝐻 = (Homa‘𝐶) | |
5 | homarcl2.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
6 | 4 | homarcl 18081 | . . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
7 | 4, 5, 6 | homaf 18083 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V)) |
8 | 7 | fdmd 6746 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → dom 𝐻 = (𝐵 × 𝐵)) |
9 | 3, 8 | eleqtrd 2840 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
10 | opelxp 5724 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵) ↔ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) | |
11 | 9, 10 | sylib 218 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 𝒫 cpw 4604 〈cop 4636 × cxp 5686 dom cdm 5688 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 Homachoma 18076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-homa 18079 |
This theorem is referenced by: homarel 18089 homa1 18090 homahom2 18091 homadm 18093 homacd 18094 arwdm 18100 arwcd 18101 coahom 18123 arwlid 18125 arwrid 18126 arwass 18127 |
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