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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 6702 | . . . 4 ⊢ (𝐹 ∈ (𝐻‘〈𝑋, 𝑌〉) → 〈𝑋, 𝑌〉 ∈ dom 𝐻) | |
2 | df-ov 7159 | . . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
3 | 1, 2 | eleq2s 2931 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 〈𝑋, 𝑌〉 ∈ dom 𝐻) |
4 | homahom.h | . . . . 5 ⊢ 𝐻 = (Homa‘𝐶) | |
5 | homarcl2.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
6 | 4 | homarcl 17288 | . . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
7 | 4, 5, 6 | homaf 17290 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V)) |
8 | 7 | fdmd 6523 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → dom 𝐻 = (𝐵 × 𝐵)) |
9 | 3, 8 | eleqtrd 2915 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
10 | opelxp 5591 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵) ↔ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) | |
11 | 9, 10 | sylib 220 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 𝒫 cpw 4539 〈cop 4573 × cxp 5553 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Homachoma 17283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-homa 17286 |
This theorem is referenced by: homarel 17296 homa1 17297 homahom2 17298 homadm 17300 homacd 17301 arwdm 17307 arwcd 17308 coahom 17330 arwlid 17332 arwrid 17333 arwass 17334 |
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