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Mirrors > Home > MPE Home > Th. List > ndxid | Structured version Visualization version GIF version |
Description: A structure component
extractor is defined by its own index. This
theorem, together with strfv 17081 below, is useful for avoiding direct
reference to the hard-coded numeric index in component extractor
definitions, such as the 1 in df-base 17089 and the ;10 in
df-ple 17158, making it easier to change should the need
arise.
For example, we can refer to a specific poset with base set π΅ and order relation πΏ using {β¨(Baseβndx), π΅β©, β¨(leβndx), πΏβ©} rather than {β¨1, π΅β©, β¨;10, πΏβ©}. The latter, while shorter to state, requires revision if we later change ;10 to some other number, and it may also be harder to remember. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.) (Proof shortened by BJ, 27-Dec-2021.) |
Ref | Expression |
---|---|
ndxarg.e | β’ πΈ = Slot π |
ndxarg.n | β’ π β β |
Ref | Expression |
---|---|
ndxid | β’ πΈ = Slot (πΈβndx) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ndxarg.e | . . . 4 β’ πΈ = Slot π | |
2 | ndxarg.n | . . . 4 β’ π β β | |
3 | 1, 2 | ndxarg 17073 | . . 3 β’ (πΈβndx) = π |
4 | 3 | eqcomi 2742 | . 2 β’ π = (πΈβndx) |
5 | sloteq 17060 | . . 3 β’ (π = (πΈβndx) β Slot π = Slot (πΈβndx)) | |
6 | 1, 5 | eqtrid 2785 | . 2 β’ (π = (πΈβndx) β πΈ = Slot (πΈβndx)) |
7 | 4, 6 | ax-mp 5 | 1 β’ πΈ = Slot (πΈβndx) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 β wcel 2107 βcfv 6497 βcn 12158 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-1cn 11114 ax-addcl 11116 |
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-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-ov 7361 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-nn 12159 df-slot 17059 df-ndx 17071 |
This theorem is referenced by: strndxid 17075 setsidvaldOLD 17077 baseid 17091 2strop 17112 2strop1 17116 resslemOLD 17128 plusgid 17165 mulrid 17180 starvid 17189 scaid 17201 vscaid 17206 ipid 17217 tsetid 17239 pleid 17253 ocid 17268 dsid 17272 unifid 17282 homid 17298 ccoid 17300 oppglemOLD 19134 mgplemOLD 19906 opprlemOLD 20060 sralemOLD 20655 zlmlemOLD 20934 znbaslemOLD 20958 opsrbaslemOLD 21467 tnglemOLD 24013 itvid 27423 lngid 27424 ttglemOLD 27862 cchhllemOLD 27878 edgfid 27981 resvlemOLD 32170 hlhilslemOLD 40448 mnringnmulrdOLD 42578 |
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