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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 6928 | . . . 4 ⊢ (𝐹 ∈ (𝐻‘〈𝑋, 𝑌〉) → 〈𝑋, 𝑌〉 ∈ dom 𝐻) | |
2 | df-ov 7417 | . . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
3 | 1, 2 | eleq2s 2846 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 〈𝑋, 𝑌〉 ∈ dom 𝐻) |
4 | homahom.h | . . . . 5 ⊢ 𝐻 = (Homa‘𝐶) | |
5 | homarcl2.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
6 | 4 | homarcl 18010 | . . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
7 | 4, 5, 6 | homaf 18012 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V)) |
8 | 7 | fdmd 6727 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → dom 𝐻 = (𝐵 × 𝐵)) |
9 | 3, 8 | eleqtrd 2830 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
10 | opelxp 5708 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵) ↔ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) | |
11 | 9, 10 | sylib 217 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3469 𝒫 cpw 4598 〈cop 4630 × cxp 5670 dom cdm 5672 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 Homachoma 18005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-homa 18008 |
This theorem is referenced by: homarel 18018 homa1 18019 homahom2 18020 homadm 18022 homacd 18023 arwdm 18029 arwcd 18030 coahom 18052 arwlid 18054 arwrid 18055 arwass 18056 |
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