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Mirrors > Home > MPE Home > Th. List > 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 | β’ (π β π½ β (SubcatβπΆ)) |
inclfusubc.s | β’ π = (πΆ βΎcat π½) |
inclfusubc.b | β’ π΅ = (Baseβπ) |
inclfusubc.f | β’ (π β πΉ = ( I βΎ π΅)) |
inclfusubc.g | β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯π½π¦)))) |
Ref | Expression |
---|---|
inclfusubc | β’ (π β πΉ(π Func πΆ)πΊ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fthfunc 17893 | . . 3 β’ (π Faith πΆ) β (π Func πΆ) | |
2 | inclfusubc.j | . . . 4 β’ (π β π½ β (SubcatβπΆ)) | |
3 | inclfusubc.s | . . . . 5 β’ π = (πΆ βΎcat π½) | |
4 | eqid 2725 | . . . . 5 β’ (idfuncβπ) = (idfuncβπ) | |
5 | 3, 4 | rescfth 17923 | . . . 4 β’ (π½ β (SubcatβπΆ) β (idfuncβπ) β (π Faith πΆ)) |
6 | 2, 5 | syl 17 | . . 3 β’ (π β (idfuncβπ) β (π Faith πΆ)) |
7 | 1, 6 | sselid 3970 | . 2 β’ (π β (idfuncβπ) β (π Func πΆ)) |
8 | df-br 5144 | . . 3 β’ (πΉ(π Func πΆ)πΊ β β¨πΉ, πΊβ© β (π Func πΆ)) | |
9 | inclfusubc.f | . . . . . 6 β’ (π β πΉ = ( I βΎ π΅)) | |
10 | inclfusubc.g | . . . . . 6 β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯π½π¦)))) | |
11 | 9, 10 | opeq12d 4877 | . . . . 5 β’ (π β β¨πΉ, πΊβ© = β¨( I βΎ π΅), (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯π½π¦)))β©) |
12 | inclfusubc.b | . . . . . . 7 β’ π΅ = (Baseβπ) | |
13 | 3, 4, 12 | idfusubc 17883 | . . . . . 6 β’ (π½ β (SubcatβπΆ) β (idfuncβπ) = β¨( I βΎ π΅), (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯π½π¦)))β©) |
14 | 2, 13 | syl 17 | . . . . 5 β’ (π β (idfuncβπ) = β¨( I βΎ π΅), (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯π½π¦)))β©) |
15 | 11, 14 | eqtr4d 2768 | . . . 4 β’ (π β β¨πΉ, πΊβ© = (idfuncβπ)) |
16 | 15 | eleq1d 2810 | . . 3 β’ (π β (β¨πΉ, πΊβ© β (π Func πΆ) β (idfuncβπ) β (π Func πΆ))) |
17 | 8, 16 | bitrid 282 | . 2 β’ (π β (πΉ(π Func πΆ)πΊ β (idfuncβπ) β (π Func πΆ))) |
18 | 7, 17 | mpbird 256 | 1 β’ (π β πΉ(π Func πΆ)πΊ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β¨cop 4630 class class class wbr 5143 I cid 5569 βΎ cres 5674 βcfv 6542 (class class class)co 7415 β cmpo 7417 Basecbs 17177 βΎcat cresc 17788 Subcatcsubc 17789 Func cfunc 17837 idfunccidfu 17838 Faith cfth 17889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-hom 17254 df-cco 17255 df-cat 17645 df-cid 17646 df-homf 17647 df-ssc 17790 df-resc 17791 df-subc 17792 df-func 17841 df-idfu 17842 df-full 17890 df-fth 17891 |
This theorem is referenced by: rngcifuestrc 20574 |
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