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Mathbox for Alexander van der Vekens |
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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 | β’ π = (πΆ βΎcat π½) |
idfusubc.i | β’ πΌ = (idfuncβπ) |
idfusubc.b | β’ π΅ = (Baseβπ) |
Ref | Expression |
---|---|
idfusubc | β’ (π½ β (SubcatβπΆ) β πΌ = β¨( I βΎ π΅), (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯π½π¦)))β©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idfusubc.s | . . 3 β’ π = (πΆ βΎcat π½) | |
2 | idfusubc.i | . . 3 β’ πΌ = (idfuncβπ) | |
3 | idfusubc.b | . . 3 β’ π΅ = (Baseβπ) | |
4 | 1, 2, 3 | idfusubc0 46639 | . 2 β’ (π½ β (SubcatβπΆ) β πΌ = β¨( I βΎ π΅), (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯(Hom βπ)π¦)))β©) |
5 | eqid 2733 | . . . . . . . 8 β’ (BaseβπΆ) = (BaseβπΆ) | |
6 | subcrcl 17763 | . . . . . . . 8 β’ (π½ β (SubcatβπΆ) β πΆ β Cat) | |
7 | id 22 | . . . . . . . . 9 β’ (π½ β (SubcatβπΆ) β π½ β (SubcatβπΆ)) | |
8 | eqidd 2734 | . . . . . . . . 9 β’ (π½ β (SubcatβπΆ) β dom dom π½ = dom dom π½) | |
9 | 7, 8 | subcfn 17791 | . . . . . . . 8 β’ (π½ β (SubcatβπΆ) β π½ Fn (dom dom π½ Γ dom dom π½)) |
10 | 7, 9, 5 | subcss1 17792 | . . . . . . . 8 β’ (π½ β (SubcatβπΆ) β dom dom π½ β (BaseβπΆ)) |
11 | 1, 5, 6, 9, 10 | reschom 17778 | . . . . . . 7 β’ (π½ β (SubcatβπΆ) β π½ = (Hom βπ)) |
12 | 11 | eqcomd 2739 | . . . . . 6 β’ (π½ β (SubcatβπΆ) β (Hom βπ) = π½) |
13 | 12 | oveqd 7426 | . . . . 5 β’ (π½ β (SubcatβπΆ) β (π₯(Hom βπ)π¦) = (π₯π½π¦)) |
14 | 13 | reseq2d 5982 | . . . 4 β’ (π½ β (SubcatβπΆ) β ( I βΎ (π₯(Hom βπ)π¦)) = ( I βΎ (π₯π½π¦))) |
15 | 14 | mpoeq3dv 7488 | . . 3 β’ (π½ β (SubcatβπΆ) β (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯(Hom βπ)π¦))) = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯π½π¦)))) |
16 | 15 | opeq2d 4881 | . 2 β’ (π½ β (SubcatβπΆ) β β¨( I βΎ π΅), (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯(Hom βπ)π¦)))β© = β¨( I βΎ π΅), (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯π½π¦)))β©) |
17 | 4, 16 | eqtrd 2773 | 1 β’ (π½ β (SubcatβπΆ) β πΌ = β¨( I βΎ π΅), (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯π½π¦)))β©) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β¨cop 4635 I cid 5574 dom cdm 5677 βΎ cres 5679 βcfv 6544 (class class class)co 7409 β cmpo 7411 Basecbs 17144 Hom chom 17208 Catccat 17608 βΎcat cresc 17755 Subcatcsubc 17756 idfunccidfu 17805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-hom 17221 df-cco 17222 df-cat 17612 df-cid 17613 df-homf 17614 df-ssc 17757 df-resc 17758 df-subc 17759 df-idfu 17809 |
This theorem is referenced by: inclfusubc 46641 |
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