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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 16271 below, is useful for avoiding direct
reference to the hard-coded numeric index in component extractor
definitions, such as the 1 in df-base 16229 and the ;10 in
df-ple 16326, 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 16248 | . . 3 ⊢ (𝐸‘ndx) = 𝑁 |
4 | 3 | eqcomi 2835 | . 2 ⊢ 𝑁 = (𝐸‘ndx) |
5 | sloteq 16228 | . . 3 ⊢ (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx)) | |
6 | 1, 5 | syl5eq 2874 | . 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 1658 ∈ wcel 2166 ‘cfv 6124 ℕcn 11351 ndxcnx 16220 Slot cslot 16222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-1cn 10311 ax-addcl 10313 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-om 7328 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-nn 11352 df-ndx 16226 df-slot 16227 |
This theorem is referenced by: strndxid 16250 setsidvald 16254 baseid 16283 resslem 16297 plusgid 16337 2strop 16345 2strop1 16348 mulrid 16357 starvid 16365 scaid 16374 vscaid 16376 ipid 16383 tsetid 16401 pleid 16408 ocid 16415 dsid 16417 unifid 16419 homid 16429 ccoid 16431 oppglem 18131 mgplem 18849 opprlem 18983 sralem 19539 opsrbaslem 19839 zlmlem 20226 znbaslem 20247 tnglem 22815 itvid 25755 lngid 25756 ttglem 26176 cchhllem 26187 edgfid 26290 resvlem 30377 hlhilslem 38014 |
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