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| Mirrors > Home > MPE Home > Th. List > idahom | Structured version Visualization version GIF version | ||
| Description: Domain and codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| Ref | Expression |
|---|---|
| idafval.i | ⊢ 𝐼 = (Ida‘𝐶) |
| idafval.b | ⊢ 𝐵 = (Base‘𝐶) |
| idafval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
| idahom.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| idahom.h | ⊢ 𝐻 = (Homa‘𝐶) |
| Ref | Expression |
|---|---|
| idahom | ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐻𝑋)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | idafval.i | . . 3 ⊢ 𝐼 = (Ida‘𝐶) | |
| 2 | idafval.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
| 3 | idafval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
| 4 | eqid 2737 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
| 5 | idahom.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 6 | 1, 2, 3, 4, 5 | idaval 18103 | . 2 ⊢ (𝜑 → (𝐼‘𝑋) = 〈𝑋, 𝑋, ((Id‘𝐶)‘𝑋)〉) |
| 7 | idahom.h | . . 3 ⊢ 𝐻 = (Homa‘𝐶) | |
| 8 | eqid 2737 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
| 9 | 2, 8, 4, 3, 5 | catidcl 17725 | . . 3 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋)) |
| 10 | 7, 2, 3, 8, 5, 5, 9 | elhomai2 18079 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑋, ((Id‘𝐶)‘𝑋)〉 ∈ (𝑋𝐻𝑋)) |
| 11 | 6, 10 | eqeltrd 2841 | 1 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐻𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 〈cotp 4634 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Hom chom 17308 Catccat 17707 Idccid 17708 Homachoma 18068 Idacida 18098 |
| 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-rmo 3380 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-ot 4635 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-riota 7388 df-ov 7434 df-cat 17711 df-cid 17712 df-homa 18071 df-ida 18100 |
| This theorem is referenced by: idadm 18106 idacd 18107 idaf 18108 arwlid 18117 arwrid 18118 |
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