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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > idfusubc0 | Structured version Visualization version GIF version |
Description: The identity functor for a subcategory is an "inclusion functor" from the subcategory into its supercategory. (Contributed by AV, 29-Mar-2020.) |
Ref | Expression |
---|---|
idfusubc.s | ⊢ 𝑆 = (𝐶 ↾cat 𝐽) |
idfusubc.i | ⊢ 𝐼 = (idfunc‘𝑆) |
idfusubc.b | ⊢ 𝐵 = (Base‘𝑆) |
Ref | Expression |
---|---|
idfusubc0 | ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐼 = 〈( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥(Hom ‘𝑆)𝑦)))〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idfusubc.i | . . 3 ⊢ 𝐼 = (idfunc‘𝑆) | |
2 | idfusubc.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
3 | idfusubc.s | . . . 4 ⊢ 𝑆 = (𝐶 ↾cat 𝐽) | |
4 | id 22 | . . . 4 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐽 ∈ (Subcat‘𝐶)) | |
5 | 3, 4 | subccat 16904 | . . 3 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝑆 ∈ Cat) |
6 | eqid 2778 | . . 3 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆) | |
7 | 1, 2, 5, 6 | idfuval 16932 | . 2 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐼 = 〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝑆)‘𝑧)))〉) |
8 | fveq2 6448 | . . . . . . 7 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → ((Hom ‘𝑆)‘𝑧) = ((Hom ‘𝑆)‘〈𝑥, 𝑦〉)) | |
9 | df-ov 6927 | . . . . . . 7 ⊢ (𝑥(Hom ‘𝑆)𝑦) = ((Hom ‘𝑆)‘〈𝑥, 𝑦〉) | |
10 | 8, 9 | syl6eqr 2832 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → ((Hom ‘𝑆)‘𝑧) = (𝑥(Hom ‘𝑆)𝑦)) |
11 | 10 | reseq2d 5644 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → ( I ↾ ((Hom ‘𝑆)‘𝑧)) = ( I ↾ (𝑥(Hom ‘𝑆)𝑦))) |
12 | 11 | mpt2mpt 7031 | . . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝑆)‘𝑧))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥(Hom ‘𝑆)𝑦))) |
13 | 12 | a1i 11 | . . 3 ⊢ (𝐽 ∈ (Subcat‘𝐶) → (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝑆)‘𝑧))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥(Hom ‘𝑆)𝑦)))) |
14 | 13 | opeq2d 4645 | . 2 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝑆)‘𝑧)))〉 = 〈( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥(Hom ‘𝑆)𝑦)))〉) |
15 | 7, 14 | eqtrd 2814 | 1 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐼 = 〈( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥(Hom ‘𝑆)𝑦)))〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 〈cop 4404 ↦ cmpt 4967 I cid 5262 × cxp 5355 ↾ cres 5359 ‘cfv 6137 (class class class)co 6924 ↦ cmpt2 6926 Basecbs 16266 Hom chom 16360 ↾cat cresc 16864 Subcatcsubc 16865 idfunccidfu 16911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-hom 16373 df-cco 16374 df-cat 16725 df-cid 16726 df-homf 16727 df-ssc 16866 df-resc 16867 df-subc 16868 df-idfu 16915 |
This theorem is referenced by: idfusubc 42895 |
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