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| Mirrors > Home > MPE Home > Th. List > homahom2 | Structured version Visualization version GIF version | ||
| Description: The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.) |
| Ref | Expression |
|---|---|
| homahom.h | ⊢ 𝐻 = (Homa‘𝐶) |
| homahom.j | ⊢ 𝐽 = (Hom ‘𝐶) |
| Ref | Expression |
|---|---|
| homahom2 | ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝐹 ∈ (𝑋𝐽𝑌)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-br 5144 | . . . 4 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 ↔ 〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌)) | |
| 2 | homahom.h | . . . . 5 ⊢ 𝐻 = (Homa‘𝐶) | |
| 3 | eqid 2737 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
| 4 | 2 | homarcl 18073 | . . . . 5 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
| 5 | homahom.j | . . . . 5 ⊢ 𝐽 = (Hom ‘𝐶) | |
| 6 | 2, 3 | homarcl2 18080 | . . . . . 6 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
| 7 | 6 | simpld 494 | . . . . 5 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶)) |
| 8 | 6 | simprd 495 | . . . . 5 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶)) |
| 9 | 2, 3, 4, 5, 7, 8 | elhoma 18077 | . . . 4 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌)))) |
| 10 | 1, 9 | sylbi 217 | . . 3 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌)))) |
| 11 | 10 | ibi 267 | . 2 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌))) |
| 12 | 11 | simprd 495 | 1 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝐹 ∈ (𝑋𝐽𝑌)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Hom chom 17308 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-homa 18071 |
| This theorem is referenced by: homahom 18084 |
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