![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2734 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
5 | idahom.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | 1, 2, 3, 4, 5 | idaval 18111 | . 2 ⊢ (𝜑 → (𝐼‘𝑋) = 〈𝑋, 𝑋, ((Id‘𝐶)‘𝑋)〉) |
7 | idahom.h | . . 3 ⊢ 𝐻 = (Homa‘𝐶) | |
8 | eqid 2734 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
9 | 2, 8, 4, 3, 5 | catidcl 17726 | . . 3 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋)) |
10 | 7, 2, 3, 8, 5, 5, 9 | elhomai2 18087 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑋, ((Id‘𝐶)‘𝑋)〉 ∈ (𝑋𝐻𝑋)) |
11 | 6, 10 | eqeltrd 2838 | 1 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐻𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 〈cotp 4638 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 Hom chom 17308 Catccat 17708 Idccid 17709 Homachoma 18076 Idacida 18106 |
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-rmo 3377 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-ot 4639 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-riota 7387 df-ov 7433 df-cat 17712 df-cid 17713 df-homa 18079 df-ida 18108 |
This theorem is referenced by: idadm 18114 idacd 18115 idaf 18116 arwlid 18125 arwrid 18126 |
Copyright terms: Public domain | W3C validator |