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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 16533 below, is useful for avoiding direct
reference to the hard-coded numeric index in component extractor
definitions, such as the 1 in df-base 16491 and the ;10 in
df-ple 16587, making it easier to change should the need
arise.
For example, we can refer to a specific poset with base set 𝐵 and order relation 𝐿 using {〈(Base‘ndx), 𝐵〉, 〈(le‘ndx), 𝐿〉} rather than {〈1, 𝐵〉, 〈;10, 𝐿〉}. 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.1 | ⊢ 𝐸 = Slot 𝑁 |
ndxarg.2 | ⊢ 𝑁 ∈ ℕ |
Ref | Expression |
---|---|
ndxid | ⊢ 𝐸 = Slot (𝐸‘ndx) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ndxarg.1 | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
2 | ndxarg.2 | . . . 4 ⊢ 𝑁 ∈ ℕ | |
3 | 1, 2 | ndxarg 16510 | . . 3 ⊢ (𝐸‘ndx) = 𝑁 |
4 | 3 | eqcomi 2832 | . 2 ⊢ 𝑁 = (𝐸‘ndx) |
5 | sloteq 16490 | . . 3 ⊢ (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx)) | |
6 | 1, 5 | syl5eq 2870 | . 2 ⊢ (𝑁 = (𝐸‘ndx) → 𝐸 = Slot (𝐸‘ndx)) |
7 | 4, 6 | ax-mp 5 | 1 ⊢ 𝐸 = Slot (𝐸‘ndx) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ‘cfv 6357 ℕcn 11640 ndxcnx 16482 Slot cslot 16484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-1cn 10597 ax-addcl 10599 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-nn 11641 df-ndx 16488 df-slot 16489 |
This theorem is referenced by: strndxid 16512 setsidvald 16516 baseid 16545 resslem 16559 plusgid 16598 2strop 16606 2strop1 16609 mulrid 16618 starvid 16626 scaid 16635 vscaid 16637 ipid 16644 tsetid 16662 pleid 16669 ocid 16676 dsid 16678 unifid 16680 homid 16690 ccoid 16692 oppglem 18480 mgplem 19246 opprlem 19380 sralem 19951 opsrbaslem 20260 zlmlem 20666 znbaslem 20687 tnglem 23251 itvid 26230 lngid 26231 ttglem 26664 cchhllem 26675 edgfid 26778 resvlem 30906 hlhilslem 39076 |
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