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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 2735 | . . 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 17722 | . 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 17720 | . . 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 7405 | . . 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 2839 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃!wreu 3376 〈cop 4637 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 Basecbs 17245 Hom chom 17309 compcco 17310 Catccat 17709 Idccid 17710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-cat 17713 df-cid 17714 |
This theorem is referenced by: oppccatid 17766 monsect 17831 sectid 17834 catsubcat 17890 fullsubc 17901 idfucl 17932 cofucl 17939 fthsect 17979 fucidcl 18022 initoid 18055 termoid 18056 idahom 18114 catcisolem 18164 xpccatid 18244 1stfcl 18253 2ndfcl 18254 prfcl 18259 evlfcl 18279 curf1cl 18285 curf2cl 18288 curfcl 18289 curfuncf 18295 uncfcurf 18296 diag12 18301 diag2 18302 curf2ndf 18304 hofcl 18316 yon12 18322 yon2 18323 yonedalem3a 18331 yonedalem3b 18336 yonedainv 18338 bj-endmnd 37301 catprs 48800 endmndlem 48804 idmon 48805 idepi 48806 upciclem3 48814 thincid 48833 functhinclem4 48844 |
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