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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 17548) with a component extractor 𝐸 (such as the base set extractor df-base 16481). By virtue of ndxid 16501, 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 16486 | . . 3 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ 𝑆 ∈ V |
4 | 1 | structfun 16491 | . 2 ⊢ Fun ◡◡𝑆 |
5 | strfv.e | . 2 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
6 | strfv.n | . . 3 ⊢ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆 | |
7 | opex 5321 | . . . 4 ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ V | |
8 | 7 | snss 4679 | . . 3 ⊢ (〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 ↔ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆) |
9 | 6, 8 | mpbir 234 | . 2 ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 |
10 | 3, 4, 5, 9 | strfv2 16522 | 1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 {csn 4525 〈cop 4531 class class class wbr 5030 ‘cfv 6324 Struct cstr 16471 ndxcnx 16472 Slot cslot 16474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-slot 16479 |
This theorem is referenced by: strfv3 16524 1strbas 16591 2strbas 16595 2strop 16596 2strbas1 16598 2strop1 16599 rngbase 16612 rngplusg 16613 rngmulr 16614 srngbase 16620 srngplusg 16621 srngmulr 16622 srnginvl 16623 lmodbase 16629 lmodplusg 16630 lmodsca 16631 lmodvsca 16632 ipsbase 16636 ipsaddg 16637 ipsmulr 16638 ipssca 16639 ipsvsca 16640 ipsip 16641 phlbase 16646 phlplusg 16647 phlsca 16648 phlvsca 16649 phlip 16650 topgrpbas 16654 topgrpplusg 16655 topgrptset 16656 otpsbas 16661 otpstset 16662 otpsle 16663 odrngbas 16672 odrngplusg 16673 odrngmulr 16674 odrngtset 16675 odrngle 16676 odrngds 16677 imassca 16784 imastset 16787 fuccofval 17221 setcbas 17330 catchomfval 17350 catccofval 17352 estrcbas 17367 ipobas 17757 ipolerval 17758 ipotset 17759 cnfldbas 20095 cnfldadd 20096 cnfldmul 20097 cnfldcj 20098 cnfldtset 20099 cnfldle 20100 cnfldds 20101 cnfldunif 20102 psrbas 20616 psrplusg 20619 psrmulr 20622 psrsca 20627 psrvscafval 20628 trkgbas 26239 trkgdist 26240 trkgitv 26241 idlsrgbas 31057 idlsrgplusg 31058 idlsrgmulr 31060 idlsrgtset 31061 algbase 40122 algaddg 40123 algmulr 40124 algsca 40125 algvsca 40126 rngchomfvalALTV 44608 rngccofvalALTV 44611 ringchomfvalALTV 44671 ringccofvalALTV 44674 |
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