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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 2610 | . . 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 16161 | . 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 16159 | . . 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 6525 | . . 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 2688 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃!wreu 2898 〈cop 4131 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 Basecbs 15695 Hom chom 15779 compcco 15780 Catccat 16148 Idccid 16149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-cat 16152 df-cid 16153 |
This theorem is referenced by: oppccatid 16202 monsect 16266 sectid 16269 cicref 16284 catsubcat 16322 fullsubc 16333 idfucl 16364 cofucl 16371 fthsect 16408 fucidcl 16448 initoid 16478 termoid 16479 idahom 16533 catcisolem 16579 xpccatid 16651 1stfcl 16660 2ndfcl 16661 prfcl 16666 evlfcl 16685 curf1cl 16691 curf2cl 16694 curfcl 16695 curfuncf 16701 uncfcurf 16702 diag12 16707 diag2 16708 curf2ndf 16710 hofcl 16722 yon12 16728 yon2 16729 yonedalem3a 16737 yonedalem3b 16742 yonedainv 16744 |
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