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Mirrors > Home > MPE Home > Th. List > strfv | Structured version Visualization version GIF version |
Description: Extract a structure component πΆ (such as the base set) from a structure π (such as a member of Poset, df-poset 18207) with a component extractor πΈ (such as the base set extractor df-base 17089). By virtue of ndxid 17074, this can be done without having to refer to the hard-coded numeric index of πΈ. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.) |
Ref | Expression |
---|---|
strfv.s | β’ π Struct π |
strfv.e | β’ πΈ = Slot (πΈβndx) |
strfv.n | β’ {β¨(πΈβndx), πΆβ©} β π |
Ref | Expression |
---|---|
strfv | β’ (πΆ β π β πΆ = (πΈβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strfv.s | . . 3 β’ π Struct π | |
2 | structex 17027 | . . 3 β’ (π Struct π β π β V) | |
3 | 1, 2 | ax-mp 5 | . 2 β’ π β V |
4 | 1 | structfun 17032 | . 2 β’ Fun β‘β‘π |
5 | strfv.e | . 2 β’ πΈ = Slot (πΈβndx) | |
6 | strfv.n | . . 3 β’ {β¨(πΈβndx), πΆβ©} β π | |
7 | opex 5422 | . . . 4 β’ β¨(πΈβndx), πΆβ© β V | |
8 | 7 | snss 4747 | . . 3 β’ (β¨(πΈβndx), πΆβ© β π β {β¨(πΈβndx), πΆβ©} β π) |
9 | 6, 8 | mpbir 230 | . 2 β’ β¨(πΈβndx), πΆβ© β π |
10 | 3, 4, 5, 9 | strfv2 17080 | 1 β’ (πΆ β π β πΆ = (πΈβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3444 β wss 3911 {csn 4587 β¨cop 4593 class class class wbr 5106 βcfv 6497 Struct cstr 17023 Slot cslot 17058 ndxcnx 17070 |
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-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-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-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 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-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 |
This theorem is referenced by: strfv3 17082 1strbas 17105 1strbasOLD 17106 2strbas 17111 2strop 17112 2strbas1 17115 2strop1 17116 rngbase 17185 rngplusg 17186 rngmulr 17187 srngbase 17196 srngplusg 17197 srngmulr 17198 srnginvl 17199 lmodbase 17212 lmodplusg 17213 lmodsca 17214 lmodvsca 17215 ipsbase 17223 ipsaddg 17224 ipsmulr 17225 ipssca 17226 ipsvsca 17227 ipsip 17228 phlbase 17233 phlplusg 17234 phlsca 17235 phlvsca 17236 phlip 17237 topgrpbas 17248 topgrpplusg 17249 topgrptset 17250 otpsbas 17263 otpstset 17264 otpsle 17265 odrngbas 17290 odrngplusg 17291 odrngmulr 17292 odrngtset 17293 odrngle 17294 odrngds 17295 imassca 17406 imastset 17409 fuccofval 17852 setcbas 17969 catchomfval 17993 catccofval 17995 estrcbas 18017 ipobas 18425 ipolerval 18426 ipotset 18427 cnfldbas 20816 cnfldadd 20817 cnfldmul 20818 cnfldcj 20819 cnfldtset 20820 cnfldle 20821 cnfldds 20822 cnfldunif 20823 psrbas 21362 psrplusg 21365 psrmulr 21368 psrsca 21373 psrvscafval 21374 trkgbas 27429 trkgdist 27430 trkgitv 27431 idlsrgbas 32294 idlsrgplusg 32295 idlsrgmulr 32297 idlsrgtset 32298 algbase 41548 algaddg 41549 algmulr 41550 algsca 41551 algvsca 41552 rngchomfvalALTV 46368 rngccofvalALTV 46371 ringchomfvalALTV 46431 ringccofvalALTV 46434 mndtcbasval 47192 |
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