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| Mirrors > Home > MPE Home > Th. List > strfv | Structured version Visualization version GIF version | ||
| Description: Extract a structure component 𝐶 (such as the base set) from a structure 𝑆 (such as a member of Poset, df-poset 18359) with a component extractor 𝐸 (such as the base set extractor df-base 17248). By virtue of ndxid 17234, this can be done without having to refer to the hard-coded numeric index of 𝐸. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| strfv.s | ⊢ 𝑆 Struct 𝑋 | 
| strfv.e | ⊢ 𝐸 = Slot (𝐸‘ndx) | 
| strfv.n | ⊢ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆 | 
| Ref | Expression | 
|---|---|
| strfv | ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | strfv.s | . . 3 ⊢ 𝑆 Struct 𝑋 | |
| 2 | structex 17187 | . . 3 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V) | |
| 3 | 1, 2 | ax-mp 5 | . 2 ⊢ 𝑆 ∈ V | 
| 4 | 1 | structfun 17192 | . 2 ⊢ Fun ◡◡𝑆 | 
| 5 | strfv.e | . 2 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
| 6 | strfv.n | . . 3 ⊢ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆 | |
| 7 | opex 5469 | . . . 4 ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ V | |
| 8 | 7 | snss 4785 | . . 3 ⊢ (〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 ↔ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆) | 
| 9 | 6, 8 | mpbir 231 | . 2 ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 | 
| 10 | 3, 4, 5, 9 | strfv2 17239 | 1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 {csn 4626 〈cop 4632 class class class wbr 5143 ‘cfv 6561 Struct cstr 17183 Slot cslot 17218 ndxcnx 17230 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 | 
| This theorem is referenced by: strfv3 17241 1strbas 17263 1strbasOLD 17264 2strbas 17268 2strop 17269 2strbas1 17272 2strop1 17273 rngbase 17343 rngplusg 17344 rngmulr 17345 srngbase 17354 srngplusg 17355 srngmulr 17356 srnginvl 17357 lmodbase 17370 lmodplusg 17371 lmodsca 17372 lmodvsca 17373 ipsbase 17381 ipsaddg 17382 ipsmulr 17383 ipssca 17384 ipsvsca 17385 ipsip 17386 phlbase 17391 phlplusg 17392 phlsca 17393 phlvsca 17394 phlip 17395 topgrpbas 17406 topgrpplusg 17407 topgrptset 17408 otpsbas 17421 otpstset 17422 otpsle 17423 odrngbas 17448 odrngplusg 17449 odrngmulr 17450 odrngtset 17451 odrngle 17452 odrngds 17453 imassca 17564 imastset 17567 fuccofval 18007 setcbas 18123 catchomfval 18147 catccofval 18149 estrcbas 18169 ipobas 18576 ipolerval 18577 ipotset 18578 cnfldbas 21368 mpocnfldadd 21369 mpocnfldmul 21371 cnfldcj 21373 cnfldtset 21374 cnfldle 21375 cnfldds 21376 cnfldunif 21377 cnfldbasOLD 21383 cnfldaddOLD 21384 cnfldmulOLD 21385 cnfldcjOLD 21386 cnfldtsetOLD 21387 cnfldleOLD 21388 cnflddsOLD 21389 cnfldunifOLD 21390 psrbas 21953 psrplusg 21956 psrmulr 21962 psrsca 21967 psrvscafval 21968 trkgbas 28453 trkgdist 28454 trkgitv 28455 idlsrgbas 33532 idlsrgplusg 33533 idlsrgmulr 33535 idlsrgtset 33536 algbase 43186 algaddg 43187 algmulr 43188 algsca 43189 algvsca 43190 rngchomfvalALTV 48183 rngccofvalALTV 48186 ringchomfvalALTV 48217 ringccofvalALTV 48220 mndtcbasval 49177 | 
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