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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 17559) with a component extractor 𝐸 (such as the base set extractor df-base 16492). By virtue of ndxid 16512, 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 16497 | . . 3 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ 𝑆 ∈ V |
4 | 1 | structfun 16502 | . 2 ⊢ Fun ◡◡𝑆 |
5 | strfv.e | . 2 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
6 | strfv.n | . . 3 ⊢ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆 | |
7 | opex 5359 | . . . 4 ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ V | |
8 | 7 | snss 4721 | . . 3 ⊢ (〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 ↔ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆) |
9 | 6, 8 | mpbir 233 | . 2 ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 |
10 | 3, 4, 5, 9 | strfv2 16533 | 1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ⊆ wss 3939 {csn 4570 〈cop 4576 class class class wbr 5069 ‘cfv 6358 Struct cstr 16482 ndxcnx 16483 Slot cslot 16485 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16488 df-slot 16490 |
This theorem is referenced by: strfv3 16535 1strbas 16602 2strbas 16606 2strop 16607 2strbas1 16609 2strop1 16610 rngbase 16623 rngplusg 16624 rngmulr 16625 srngbase 16631 srngplusg 16632 srngmulr 16633 srnginvl 16634 lmodbase 16640 lmodplusg 16641 lmodsca 16642 lmodvsca 16643 ipsbase 16647 ipsaddg 16648 ipsmulr 16649 ipssca 16650 ipsvsca 16651 ipsip 16652 phlbase 16657 phlplusg 16658 phlsca 16659 phlvsca 16660 phlip 16661 topgrpbas 16665 topgrpplusg 16666 topgrptset 16667 otpsbas 16672 otpstset 16673 otpsle 16674 odrngbas 16683 odrngplusg 16684 odrngmulr 16685 odrngtset 16686 odrngle 16687 odrngds 16688 imassca 16795 imastset 16798 fuccofval 17232 setcbas 17341 catchomfval 17361 catccofval 17363 estrcbas 17378 ipobas 17768 ipolerval 17769 ipotset 17770 psrbas 20161 psrplusg 20164 psrmulr 20167 psrsca 20172 psrvscafval 20173 cnfldbas 20552 cnfldadd 20553 cnfldmul 20554 cnfldcj 20555 cnfldtset 20556 cnfldle 20557 cnfldds 20558 cnfldunif 20559 trkgbas 26234 trkgdist 26235 trkgitv 26236 algbase 39784 algaddg 39785 algmulr 39786 algsca 39787 algvsca 39788 rngchomfvalALTV 44262 rngccofvalALTV 44265 ringchomfvalALTV 44325 ringccofvalALTV 44328 |
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