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Mirrors > Home > MPE Home > Th. List > rescbasOLD | Structured version Visualization version GIF version |
Description: Obsolete version of rescbas 17719 as of 18-Oct-2024. Base set of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rescbas.d | β’ π· = (πΆ βΎcat π») |
rescbas.b | β’ π΅ = (BaseβπΆ) |
rescbas.c | β’ (π β πΆ β π) |
rescbas.h | β’ (π β π» Fn (π Γ π)) |
rescbas.s | β’ (π β π β π΅) |
Ref | Expression |
---|---|
rescbasOLD | β’ (π β π = (Baseβπ·)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baseid 17093 | . . 3 β’ Base = Slot (Baseβndx) | |
2 | 1re 11162 | . . . . 5 β’ 1 β β | |
3 | 1nn 12171 | . . . . . 6 β’ 1 β β | |
4 | 4nn0 12439 | . . . . . 6 β’ 4 β β0 | |
5 | 1nn0 12436 | . . . . . 6 β’ 1 β β0 | |
6 | 1lt10 12764 | . . . . . 6 β’ 1 < ;10 | |
7 | 3, 4, 5, 6 | declti 12663 | . . . . 5 β’ 1 < ;14 |
8 | 2, 7 | ltneii 11275 | . . . 4 β’ 1 β ;14 |
9 | basendx 17099 | . . . . 5 β’ (Baseβndx) = 1 | |
10 | homndx 17299 | . . . . 5 β’ (Hom βndx) = ;14 | |
11 | 9, 10 | neeq12i 3011 | . . . 4 β’ ((Baseβndx) β (Hom βndx) β 1 β ;14) |
12 | 8, 11 | mpbir 230 | . . 3 β’ (Baseβndx) β (Hom βndx) |
13 | 1, 12 | setsnid 17088 | . 2 β’ (Baseβ(πΆ βΎs π)) = (Baseβ((πΆ βΎs π) sSet β¨(Hom βndx), π»β©)) |
14 | rescbas.s | . . 3 β’ (π β π β π΅) | |
15 | eqid 2737 | . . . 4 β’ (πΆ βΎs π) = (πΆ βΎs π) | |
16 | rescbas.b | . . . 4 β’ π΅ = (BaseβπΆ) | |
17 | 15, 16 | ressbas2 17127 | . . 3 β’ (π β π΅ β π = (Baseβ(πΆ βΎs π))) |
18 | 14, 17 | syl 17 | . 2 β’ (π β π = (Baseβ(πΆ βΎs π))) |
19 | rescbas.d | . . . 4 β’ π· = (πΆ βΎcat π») | |
20 | rescbas.c | . . . 4 β’ (π β πΆ β π) | |
21 | 16 | fvexi 6861 | . . . . . 6 β’ π΅ β V |
22 | 21 | ssex 5283 | . . . . 5 β’ (π β π΅ β π β V) |
23 | 14, 22 | syl 17 | . . . 4 β’ (π β π β V) |
24 | rescbas.h | . . . 4 β’ (π β π» Fn (π Γ π)) | |
25 | 19, 20, 23, 24 | rescval2 17718 | . . 3 β’ (π β π· = ((πΆ βΎs π) sSet β¨(Hom βndx), π»β©)) |
26 | 25 | fveq2d 6851 | . 2 β’ (π β (Baseβπ·) = (Baseβ((πΆ βΎs π) sSet β¨(Hom βndx), π»β©))) |
27 | 13, 18, 26 | 3eqtr4a 2803 | 1 β’ (π β π = (Baseβπ·)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wne 2944 Vcvv 3448 β wss 3915 β¨cop 4597 Γ cxp 5636 Fn wfn 6496 βcfv 6501 (class class class)co 7362 1c1 11059 4c4 12217 ;cdc 12625 sSet csts 17042 ndxcnx 17072 Basecbs 17090 βΎs cress 17119 Hom chom 17151 βΎcat cresc 17698 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-hom 17164 df-resc 17701 |
This theorem is referenced by: (None) |
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