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| Mirrors > Home > MPE Home > Th. List > catidcl | Structured version Visualization version GIF version | ||
| Description: Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| catidcl.b | ⊢ 𝐵 = (Base‘𝐶) | 
| catidcl.h | ⊢ 𝐻 = (Hom ‘𝐶) | 
| catidcl.i | ⊢ 1 = (Id‘𝐶) | 
| catidcl.c | ⊢ (𝜑 → 𝐶 ∈ Cat) | 
| catidcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| Ref | Expression | 
|---|---|
| catidcl | ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | catidcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
| 2 | catidcl.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
| 3 | eqid 2736 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
| 4 | catidcl.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
| 5 | catidcl.i | . . 3 ⊢ 1 = (Id‘𝐶) | |
| 6 | catidcl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 7 | 1, 2, 3, 4, 5, 6 | cidval 17721 | . 2 ⊢ (𝜑 → ( 1 ‘𝑋) = (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉(comp‘𝐶)𝑦)𝑔) = 𝑓))) | 
| 8 | 1, 2, 3, 4, 6 | catideu 17719 | . . 3 ⊢ (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉(comp‘𝐶)𝑦)𝑔) = 𝑓)) | 
| 9 | riotacl 7406 | . . 3 ⊢ (∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉(comp‘𝐶)𝑦)𝑔) = 𝑓) → (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉(comp‘𝐶)𝑦)𝑔) = 𝑓)) ∈ (𝑋𝐻𝑋)) | |
| 10 | 8, 9 | syl 17 | . 2 ⊢ (𝜑 → (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉(comp‘𝐶)𝑦)𝑔) = 𝑓)) ∈ (𝑋𝐻𝑋)) | 
| 11 | 7, 10 | eqeltrd 2840 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃!wreu 3377 〈cop 4631 ‘cfv 6560 ℩crio 7388 (class class class)co 7432 Basecbs 17248 Hom chom 17309 compcco 17310 Catccat 17708 Idccid 17709 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-cat 17712 df-cid 17713 | 
| This theorem is referenced by: oppccatid 17763 monsect 17828 sectid 17831 catsubcat 17885 fullsubc 17896 idfucl 17927 cofucl 17934 fthsect 17973 fucidcl 18014 initoid 18047 termoid 18048 idahom 18106 catcisolem 18156 xpccatid 18234 1stfcl 18243 2ndfcl 18244 prfcl 18249 evlfcl 18268 curf1cl 18274 curf2cl 18277 curfcl 18278 curfuncf 18284 uncfcurf 18285 diag12 18290 diag2 18291 curf2ndf 18293 hofcl 18305 yon12 18311 yon2 18312 yonedalem3a 18320 yonedalem3b 18325 yonedainv 18327 bj-endmnd 37320 catprs 48915 endmndlem 48919 idmon 48923 idepi 48924 upciclem3 48943 swapfid 49003 tposcurf12 49016 tposcurf2 49018 fucoid 49066 thincid 49106 functhinclem4 49121 termchomn0 49155 idfudiag1 49183 termcarweu 49186 arweutermc 49188 | 
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