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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > inclfusubc | Structured version Visualization version GIF version |
Description: The "inclusion functor" from a subcategory of a category into the category itself. (Contributed by AV, 30-Mar-2020.) |
Ref | Expression |
---|---|
inclfusubc.j | ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) |
inclfusubc.s | ⊢ 𝑆 = (𝐶 ↾cat 𝐽) |
inclfusubc.b | ⊢ 𝐵 = (Base‘𝑆) |
inclfusubc.f | ⊢ (𝜑 → 𝐹 = ( I ↾ 𝐵)) |
inclfusubc.g | ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))) |
Ref | Expression |
---|---|
inclfusubc | ⊢ (𝜑 → 𝐹(𝑆 Func 𝐶)𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fthfunc 16926 | . . 3 ⊢ (𝑆 Faith 𝐶) ⊆ (𝑆 Func 𝐶) | |
2 | inclfusubc.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | |
3 | inclfusubc.s | . . . . 5 ⊢ 𝑆 = (𝐶 ↾cat 𝐽) | |
4 | eqid 2825 | . . . . 5 ⊢ (idfunc‘𝑆) = (idfunc‘𝑆) | |
5 | 3, 4 | rescfth 16956 | . . . 4 ⊢ (𝐽 ∈ (Subcat‘𝐶) → (idfunc‘𝑆) ∈ (𝑆 Faith 𝐶)) |
6 | 2, 5 | syl 17 | . . 3 ⊢ (𝜑 → (idfunc‘𝑆) ∈ (𝑆 Faith 𝐶)) |
7 | 1, 6 | sseldi 3825 | . 2 ⊢ (𝜑 → (idfunc‘𝑆) ∈ (𝑆 Func 𝐶)) |
8 | df-br 4876 | . . 3 ⊢ (𝐹(𝑆 Func 𝐶)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝑆 Func 𝐶)) | |
9 | inclfusubc.f | . . . . . 6 ⊢ (𝜑 → 𝐹 = ( I ↾ 𝐵)) | |
10 | inclfusubc.g | . . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))) | |
11 | 9, 10 | opeq12d 4633 | . . . . 5 ⊢ (𝜑 → 〈𝐹, 𝐺〉 = 〈( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))〉) |
12 | inclfusubc.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑆) | |
13 | 3, 4, 12 | idfusubc 42727 | . . . . . 6 ⊢ (𝐽 ∈ (Subcat‘𝐶) → (idfunc‘𝑆) = 〈( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))〉) |
14 | 2, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → (idfunc‘𝑆) = 〈( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))〉) |
15 | 11, 14 | eqtr4d 2864 | . . . 4 ⊢ (𝜑 → 〈𝐹, 𝐺〉 = (idfunc‘𝑆)) |
16 | 15 | eleq1d 2891 | . . 3 ⊢ (𝜑 → (〈𝐹, 𝐺〉 ∈ (𝑆 Func 𝐶) ↔ (idfunc‘𝑆) ∈ (𝑆 Func 𝐶))) |
17 | 8, 16 | syl5bb 275 | . 2 ⊢ (𝜑 → (𝐹(𝑆 Func 𝐶)𝐺 ↔ (idfunc‘𝑆) ∈ (𝑆 Func 𝐶))) |
18 | 7, 17 | mpbird 249 | 1 ⊢ (𝜑 → 𝐹(𝑆 Func 𝐶)𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 〈cop 4405 class class class wbr 4875 I cid 5251 ↾ cres 5348 ‘cfv 6127 (class class class)co 6910 ↦ cmpt2 6912 Basecbs 16229 ↾cat cresc 16827 Subcatcsubc 16828 Func cfunc 16873 idfunccidfu 16874 Faith cfth 16922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-map 8129 df-pm 8130 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-hom 16336 df-cco 16337 df-cat 16688 df-cid 16689 df-homf 16690 df-ssc 16829 df-resc 16830 df-subc 16831 df-func 16877 df-idfu 16878 df-full 16923 df-fth 16924 |
This theorem is referenced by: rngcifuestrc 42858 |
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