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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 15735 below, is useful for avoiding direct
reference to the hard-coded numeric index in component extractor
definitions, such as the 1 in df-base 15700 and the ;10 in
df-ple 15788, 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.) |
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 | . 2 ⊢ 𝐸 = Slot 𝑁 | |
2 | df-slot 15699 | . . 3 ⊢ Slot (𝐸‘ndx) = (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx))) | |
3 | df-slot 15699 | . . . 4 ⊢ Slot 𝑁 = (𝑥 ∈ V ↦ (𝑥‘𝑁)) | |
4 | ndxarg.2 | . . . . . . 7 ⊢ 𝑁 ∈ ℕ | |
5 | 1, 4 | ndxarg 15715 | . . . . . 6 ⊢ (𝐸‘ndx) = 𝑁 |
6 | 5 | fveq2i 6106 | . . . . 5 ⊢ (𝑥‘(𝐸‘ndx)) = (𝑥‘𝑁) |
7 | 6 | mpteq2i 4669 | . . . 4 ⊢ (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx))) = (𝑥 ∈ V ↦ (𝑥‘𝑁)) |
8 | 3, 7 | eqtr4i 2635 | . . 3 ⊢ Slot 𝑁 = (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx))) |
9 | 2, 8 | eqtr4i 2635 | . 2 ⊢ Slot (𝐸‘ndx) = Slot 𝑁 |
10 | 1, 9 | eqtr4i 2635 | 1 ⊢ 𝐸 = Slot (𝐸‘ndx) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 ↦ cmpt 4643 ‘cfv 5804 ℕcn 10897 ndxcnx 15692 Slot cslot 15694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-ndx 15698 df-slot 15699 |
This theorem is referenced by: strndxid 15717 setsidvald 15721 baseid 15747 resslem 15760 plusgid 15804 2strop 15811 2strop1 15814 mulrid 15822 starvid 15828 scaid 15837 vscaid 15839 ipid 15846 tsetid 15864 pleid 15872 pleidOLD 15873 ocid 15884 dsid 15886 unifid 15888 homid 15898 ccoid 15900 oppglem 17603 mgplem 18317 opprlem 18451 sralem 18998 opsrbaslem 19298 opsrbaslemOLD 19299 zlmlem 19684 znbaslem 19705 znbaslemOLD 19706 tnglem 22254 itvid 25141 lngid 25142 ttglem 25556 cchhllem 25567 struct2griedg 25705 uhgrstrrepe 25745 resvlem 29162 hlhilslem 36248 |
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