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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 17169 | . . 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 2798 | . . . . 5 ⊢ (idfunc‘𝑆) = (idfunc‘𝑆) | |
5 | 3, 4 | rescfth 17199 | . . . 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 3913 | . 2 ⊢ (𝜑 → (idfunc‘𝑆) ∈ (𝑆 Func 𝐶)) |
8 | df-br 5031 | . . 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 4773 | . . . . 5 ⊢ (𝜑 → 〈𝐹, 𝐺〉 = 〈( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))〉) |
12 | inclfusubc.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑆) | |
13 | 3, 4, 12 | idfusubc 44490 | . . . . . 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 2836 | . . . 4 ⊢ (𝜑 → 〈𝐹, 𝐺〉 = (idfunc‘𝑆)) |
16 | 15 | eleq1d 2874 | . . 3 ⊢ (𝜑 → (〈𝐹, 𝐺〉 ∈ (𝑆 Func 𝐶) ↔ (idfunc‘𝑆) ∈ (𝑆 Func 𝐶))) |
17 | 8, 16 | syl5bb 286 | . 2 ⊢ (𝜑 → (𝐹(𝑆 Func 𝐶)𝐺 ↔ (idfunc‘𝑆) ∈ (𝑆 Func 𝐶))) |
18 | 7, 17 | mpbird 260 | 1 ⊢ (𝜑 → 𝐹(𝑆 Func 𝐶)𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 〈cop 4531 class class class wbr 5030 I cid 5424 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 Basecbs 16475 ↾cat cresc 17070 Subcatcsubc 17071 Func cfunc 17116 idfunccidfu 17117 Faith cfth 17165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 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 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-hom 16581 df-cco 16582 df-cat 16931 df-cid 16932 df-homf 16933 df-ssc 17072 df-resc 17073 df-subc 17074 df-func 17120 df-idfu 17121 df-full 17166 df-fth 17167 |
This theorem is referenced by: rngcifuestrc 44621 |
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