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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 6525 | . . . 4 ⊢ (𝐹 ∈ (𝐻‘〈𝑋, 𝑌〉) → 〈𝑋, 𝑌〉 ∈ dom 𝐻) | |
2 | df-ov 6973 | . . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
3 | 1, 2 | eleq2s 2878 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 〈𝑋, 𝑌〉 ∈ dom 𝐻) |
4 | homahom.h | . . . . 5 ⊢ 𝐻 = (Homa‘𝐶) | |
5 | homarcl2.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
6 | 4 | homarcl 17136 | . . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
7 | 4, 5, 6 | homaf 17138 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V)) |
8 | 7 | fdmd 6347 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → dom 𝐻 = (𝐵 × 𝐵)) |
9 | 3, 8 | eleqtrd 2862 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
10 | opelxp 5436 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵) ↔ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) | |
11 | 9, 10 | sylib 210 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 Vcvv 3409 𝒫 cpw 4416 〈cop 4441 × cxp 5398 dom cdm 5400 ‘cfv 6182 (class class class)co 6970 Basecbs 16329 Homachoma 17131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-homa 17134 |
This theorem is referenced by: homarel 17144 homa1 17145 homahom2 17146 homadm 17148 homacd 17149 arwdm 17155 arwcd 17156 coahom 17178 arwlid 17180 arwrid 17181 arwass 17182 |
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