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Mirrors > Home > ILE Home > Th. List > ndxid | GIF version |
Description: A structure component
extractor is defined by its own index. This
theorem, together with strslfv 12008 below, is useful for avoiding direct
reference to the hard-coded numeric index in component extractor
definitions, such as the 1 in df-base 11970, making it easier to change
should the need arise.
(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 11987 | . . 3 ⊢ (𝐸‘ndx) = 𝑁 |
4 | 3 | eqcomi 2143 | . 2 ⊢ 𝑁 = (𝐸‘ndx) |
5 | sloteq 11969 | . . 3 ⊢ (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx)) | |
6 | 1, 5 | syl5eq 2184 | . 2 ⊢ (𝑁 = (𝐸‘ndx) → 𝐸 = Slot (𝐸‘ndx)) |
7 | 4, 6 | ax-mp 5 | 1 ⊢ 𝐸 = Slot (𝐸‘ndx) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 ‘cfv 5123 ℕcn 8725 ndxcnx 11961 Slot cslot 11963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-cnex 7716 ax-resscn 7717 ax-1re 7719 ax-addrcl 7722 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fv 5131 df-inn 8726 df-ndx 11967 df-slot 11968 |
This theorem is referenced by: ndxslid 11989 strndxid 11992 baseid 12017 plusgid 12058 mulrid 12075 starvid 12084 scaid 12092 vscaid 12095 ipid 12103 tsetid 12113 pleid 12120 dsid 12123 |
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