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| 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 18070 | . . . 4 ⊢ coda = (2nd ∘ 1st ) | |
| 2 | 1 | fveq1i 6907 | . . 3 ⊢ (coda‘𝐹) = ((2nd ∘ 1st )‘𝐹) |
| 3 | fo1st 8034 | . . . . 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 3501 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V) | |
| 7 | fvco3 7008 | . . . 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 2789 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = (2nd ‘(1st ‘𝐹))) |
| 10 | homahom.h | . . . . . 6 ⊢ 𝐻 = (Homa‘𝐶) | |
| 11 | 10 | homarel 18081 | . . . . 5 ⊢ Rel (𝑋𝐻𝑌) |
| 12 | 1st2ndbr 8067 | . . . . 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 18082 | . . . 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 2737 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
| 18 | 10, 17 | homarcl2 18080 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
| 19 | op2ndg 8027 | . . 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 2781 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = 𝑌) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 class class class wbr 5143 ∘ ccom 5689 Rel wrel 5690 ⟶wf 6557 –onto→wfo 6559 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 Basecbs 17247 codaccoda 18066 Homachoma 18068 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-1st 8014 df-2nd 8015 df-coda 18070 df-homa 18071 |
| This theorem is referenced by: arwhoma 18090 idacd 18107 homdmcoa 18112 coaval 18113 |
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