![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > homacd | Structured version Visualization version GIF version |
Description: The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homahom.h | ⊢ 𝐻 = (Homa‘𝐶) |
Ref | Expression |
---|---|
homacd | ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-coda 18078 | . . . 4 ⊢ coda = (2nd ∘ 1st ) | |
2 | 1 | fveq1i 6907 | . . 3 ⊢ (coda‘𝐹) = ((2nd ∘ 1st )‘𝐹) |
3 | fo1st 8032 | . . . . 5 ⊢ 1st :V–onto→V | |
4 | fof 6820 | . . . . 5 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ 1st :V⟶V |
6 | elex 3498 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V) | |
7 | fvco3 7007 | . . . 4 ⊢ ((1st :V⟶V ∧ 𝐹 ∈ V) → ((2nd ∘ 1st )‘𝐹) = (2nd ‘(1st ‘𝐹))) | |
8 | 5, 6, 7 | sylancr 587 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ((2nd ∘ 1st )‘𝐹) = (2nd ‘(1st ‘𝐹))) |
9 | 2, 8 | eqtrid 2786 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = (2nd ‘(1st ‘𝐹))) |
10 | homahom.h | . . . . . 6 ⊢ 𝐻 = (Homa‘𝐶) | |
11 | 10 | homarel 18089 | . . . . 5 ⊢ Rel (𝑋𝐻𝑌) |
12 | 1st2ndbr 8065 | . . . . 5 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹)) | |
13 | 11, 12 | mpan 690 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹)) |
14 | 10 | homa1 18090 | . . . 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 6910 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘(1st ‘𝐹)) = (2nd ‘〈𝑋, 𝑌〉)) |
17 | eqid 2734 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
18 | 10, 17 | homarcl2 18088 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
19 | op2ndg 8025 | . . 3 ⊢ ((𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | |
20 | 18, 19 | syl 17 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
21 | 9, 16, 20 | 3eqtrd 2778 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 〈cop 4636 class class class wbr 5147 ∘ ccom 5692 Rel wrel 5693 ⟶wf 6558 –onto→wfo 6560 ‘cfv 6562 (class class class)co 7430 1st c1st 8010 2nd c2nd 8011 Basecbs 17244 codaccoda 18074 Homachoma 18076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-1st 8012 df-2nd 8013 df-coda 18078 df-homa 18079 |
This theorem is referenced by: arwhoma 18098 idacd 18115 homdmcoa 18120 coaval 18121 |
Copyright terms: Public domain | W3C validator |