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 16567 | . . 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 2622 | . . . . 5 ⊢ (idfunc‘𝑆) = (idfunc‘𝑆) | |
5 | 3, 4 | rescfth 16597 | . . . 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 3601 | . 2 ⊢ (𝜑 → (idfunc‘𝑆) ∈ (𝑆 Func 𝐶)) |
8 | df-br 4654 | . . 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 4410 | . . . . 5 ⊢ (𝜑 → 〈𝐹, 𝐺〉 = 〈( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))〉) |
12 | inclfusubc.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑆) | |
13 | 3, 4, 12 | idfusubc 41866 | . . . . . 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 2659 | . . . 4 ⊢ (𝜑 → 〈𝐹, 𝐺〉 = (idfunc‘𝑆)) |
16 | 15 | eleq1d 2686 | . . 3 ⊢ (𝜑 → (〈𝐹, 𝐺〉 ∈ (𝑆 Func 𝐶) ↔ (idfunc‘𝑆) ∈ (𝑆 Func 𝐶))) |
17 | 8, 16 | syl5bb 272 | . 2 ⊢ (𝜑 → (𝐹(𝑆 Func 𝐶)𝐺 ↔ (idfunc‘𝑆) ∈ (𝑆 Func 𝐶))) |
18 | 7, 17 | mpbird 247 | 1 ⊢ (𝜑 → 𝐹(𝑆 Func 𝐶)𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 〈cop 4183 class class class wbr 4653 I cid 5023 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 Basecbs 15857 ↾cat cresc 16468 Subcatcsubc 16469 Func cfunc 16514 idfunccidfu 16515 Faith cfth 16563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-hom 15966 df-cco 15967 df-cat 16329 df-cid 16330 df-homf 16331 df-ssc 16470 df-resc 16471 df-subc 16472 df-func 16518 df-idfu 16519 df-full 16564 df-fth 16565 |
This theorem is referenced by: rngcifuestrc 41997 |
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