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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 | 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 17799 | . . 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 2733 | . . . . 5 β’ (idfuncβπ) = (idfuncβπ) | |
5 | 3, 4 | rescfth 17829 | . . . 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 3943 | . 2 β’ (π β (idfuncβπ) β (π Func πΆ)) |
8 | df-br 5107 | . . 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 4839 | . . . . 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 46250 | . . . . . 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 2776 | . . . 4 β’ (π β β¨πΉ, πΊβ© = (idfuncβπ)) |
16 | 15 | eleq1d 2819 | . . 3 β’ (π β (β¨πΉ, πΊβ© β (π Func πΆ) β (idfuncβπ) β (π Func πΆ))) |
17 | 8, 16 | bitrid 283 | . 2 β’ (π β (πΉ(π Func πΆ)πΊ β (idfuncβπ) β (π Func πΆ))) |
18 | 7, 17 | mpbird 257 | 1 β’ (π β πΉ(π Func πΆ)πΊ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β¨cop 4593 class class class wbr 5106 I cid 5531 βΎ cres 5636 βcfv 6497 (class class class)co 7358 β cmpo 7360 Basecbs 17088 βΎcat cresc 17696 Subcatcsubc 17697 Func cfunc 17745 idfunccidfu 17746 Faith cfth 17795 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-map 8770 df-pm 8771 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-hom 17162 df-cco 17163 df-cat 17553 df-cid 17554 df-homf 17555 df-ssc 17698 df-resc 17699 df-subc 17700 df-func 17749 df-idfu 17750 df-full 17796 df-fth 17797 |
This theorem is referenced by: rngcifuestrc 46381 |
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