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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 17284 | . . . 4 ⊢ doma = (1st ∘ 1st ) | |
2 | 1 | fveq1i 6671 | . . 3 ⊢ (doma‘𝐹) = ((1st ∘ 1st )‘𝐹) |
3 | fo1st 7709 | . . . . 5 ⊢ 1st :V–onto→V | |
4 | fof 6590 | . . . . 5 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ 1st :V⟶V |
6 | elex 3512 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V) | |
7 | fvco3 6760 | . . . 4 ⊢ ((1st :V⟶V ∧ 𝐹 ∈ V) → ((1st ∘ 1st )‘𝐹) = (1st ‘(1st ‘𝐹))) | |
8 | 5, 6, 7 | sylancr 589 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ((1st ∘ 1st )‘𝐹) = (1st ‘(1st ‘𝐹))) |
9 | 2, 8 | syl5eq 2868 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (doma‘𝐹) = (1st ‘(1st ‘𝐹))) |
10 | homahom.h | . . . . . 6 ⊢ 𝐻 = (Homa‘𝐶) | |
11 | 10 | homarel 17296 | . . . . 5 ⊢ Rel (𝑋𝐻𝑌) |
12 | 1st2ndbr 7741 | . . . . 5 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹)) | |
13 | 11, 12 | mpan 688 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹)) |
14 | 10 | homa1 17297 | . . . 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 6674 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘(1st ‘𝐹)) = (1st ‘〈𝑋, 𝑌〉)) |
17 | eqid 2821 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
18 | 10, 17 | homarcl2 17295 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
19 | op1stg 7701 | . . 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 2860 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (doma‘𝐹) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 〈cop 4573 class class class wbr 5066 ∘ ccom 5559 Rel wrel 5560 ⟶wf 6351 –onto→wfo 6353 ‘cfv 6355 (class class class)co 7156 1st c1st 7687 2nd c2nd 7688 Basecbs 16483 domacdoma 17280 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-1st 7689 df-2nd 7690 df-doma 17284 df-homa 17286 |
This theorem is referenced by: arwhoma 17305 idadm 17321 homdmcoa 17327 coaval 17328 |
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