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Mirrors > Home > MPE Home > Th. List > homadm | Structured version Visualization version GIF version |
Description: The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homahom.h | ⊢ 𝐻 = (Homa‘𝐶) |
Ref | Expression |
---|---|
homadm | ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (doma‘𝐹) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-doma 16895 | . . . 4 ⊢ doma = (1st ∘ 1st ) | |
2 | 1 | fveq1i 6354 | . . 3 ⊢ (doma‘𝐹) = ((1st ∘ 1st )‘𝐹) |
3 | fo1st 7354 | . . . . 5 ⊢ 1st :V–onto→V | |
4 | fof 6277 | . . . . 5 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ 1st :V⟶V |
6 | elex 3352 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V) | |
7 | fvco3 6438 | . . . 4 ⊢ ((1st :V⟶V ∧ 𝐹 ∈ V) → ((1st ∘ 1st )‘𝐹) = (1st ‘(1st ‘𝐹))) | |
8 | 5, 6, 7 | sylancr 698 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ((1st ∘ 1st )‘𝐹) = (1st ‘(1st ‘𝐹))) |
9 | 2, 8 | syl5eq 2806 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (doma‘𝐹) = (1st ‘(1st ‘𝐹))) |
10 | homahom.h | . . . . . 6 ⊢ 𝐻 = (Homa‘𝐶) | |
11 | 10 | homarel 16907 | . . . . 5 ⊢ Rel (𝑋𝐻𝑌) |
12 | 1st2ndbr 7385 | . . . . 5 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹)) | |
13 | 11, 12 | mpan 708 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹)) |
14 | 10 | homa1 16908 | . . . 4 ⊢ ((1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹) → (1st ‘𝐹) = 〈𝑋, 𝑌〉) |
15 | 13, 14 | syl 17 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘𝐹) = 〈𝑋, 𝑌〉) |
16 | 15 | fveq2d 6357 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘(1st ‘𝐹)) = (1st ‘〈𝑋, 𝑌〉)) |
17 | eqid 2760 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
18 | 10, 17 | homarcl2 16906 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
19 | op1stg 7346 | . . 3 ⊢ ((𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)) → (1st ‘〈𝑋, 𝑌〉) = 𝑋) | |
20 | 18, 19 | syl 17 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘〈𝑋, 𝑌〉) = 𝑋) |
21 | 9, 16, 20 | 3eqtrd 2798 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (doma‘𝐹) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 〈cop 4327 class class class wbr 4804 ∘ ccom 5270 Rel wrel 5271 ⟶wf 6045 –onto→wfo 6047 ‘cfv 6049 (class class class)co 6814 1st c1st 7332 2nd c2nd 7333 Basecbs 16079 domacdoma 16891 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-1st 7334 df-2nd 7335 df-doma 16895 df-homa 16897 |
This theorem is referenced by: arwhoma 16916 idadm 16932 homdmcoa 16938 coaval 16939 |
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