Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > idfusubc | 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 |
---|---|
idfusubc | ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐼 = 〈( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idfusubc.s | . . 3 ⊢ 𝑆 = (𝐶 ↾cat 𝐽) | |
2 | idfusubc.i | . . 3 ⊢ 𝐼 = (idfunc‘𝑆) | |
3 | idfusubc.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
4 | 1, 2, 3 | idfusubc0 44064 | . 2 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐼 = 〈( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥(Hom ‘𝑆)𝑦)))〉) |
5 | eqid 2818 | . . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
6 | subcrcl 17074 | . . . . . . . 8 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat) | |
7 | id 22 | . . . . . . . . 9 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐽 ∈ (Subcat‘𝐶)) | |
8 | eqidd 2819 | . . . . . . . . 9 ⊢ (𝐽 ∈ (Subcat‘𝐶) → dom dom 𝐽 = dom dom 𝐽) | |
9 | 7, 8 | subcfn 17099 | . . . . . . . 8 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽)) |
10 | 7, 9, 5 | subcss1 17100 | . . . . . . . 8 ⊢ (𝐽 ∈ (Subcat‘𝐶) → dom dom 𝐽 ⊆ (Base‘𝐶)) |
11 | 1, 5, 6, 9, 10 | reschom 17088 | . . . . . . 7 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐽 = (Hom ‘𝑆)) |
12 | 11 | eqcomd 2824 | . . . . . 6 ⊢ (𝐽 ∈ (Subcat‘𝐶) → (Hom ‘𝑆) = 𝐽) |
13 | 12 | oveqd 7162 | . . . . 5 ⊢ (𝐽 ∈ (Subcat‘𝐶) → (𝑥(Hom ‘𝑆)𝑦) = (𝑥𝐽𝑦)) |
14 | 13 | reseq2d 5846 | . . . 4 ⊢ (𝐽 ∈ (Subcat‘𝐶) → ( I ↾ (𝑥(Hom ‘𝑆)𝑦)) = ( I ↾ (𝑥𝐽𝑦))) |
15 | 14 | mpoeq3dv 7222 | . . 3 ⊢ (𝐽 ∈ (Subcat‘𝐶) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥(Hom ‘𝑆)𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))) |
16 | 15 | opeq2d 4802 | . 2 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 〈( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥(Hom ‘𝑆)𝑦)))〉 = 〈( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))〉) |
17 | 4, 16 | eqtrd 2853 | 1 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐼 = 〈( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 〈cop 4563 I cid 5452 dom cdm 5548 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 Basecbs 16471 Hom chom 16564 Catccat 16923 ↾cat cresc 17066 Subcatcsubc 17067 idfunccidfu 17113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-hom 16577 df-cco 16578 df-cat 16927 df-cid 16928 df-homf 16929 df-ssc 17068 df-resc 17069 df-subc 17070 df-idfu 17117 |
This theorem is referenced by: inclfusubc 44066 |
Copyright terms: Public domain | W3C validator |