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Mirrors > Home > ILE Home > Th. List > basendxnn | GIF version |
Description: The index value of the base set extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 23-Sep-2020.) |
Ref | Expression |
---|---|
basendxnn | ⊢ (Base‘ndx) ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-base 11968 | . . 3 ⊢ Base = Slot 1 | |
2 | 1nn 8734 | . . 3 ⊢ 1 ∈ ℕ | |
3 | 1, 2 | ndxarg 11985 | . 2 ⊢ (Base‘ndx) = 1 |
4 | 3, 2 | eqeltri 2212 | 1 ⊢ (Base‘ndx) ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 ‘cfv 5123 1c1 7624 ℕcn 8723 ndxcnx 11959 Basecbs 11962 |
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 7714 ax-resscn 7715 ax-1re 7717 ax-addrcl 7720 |
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 8724 df-ndx 11965 df-slot 11966 df-base 11968 |
This theorem is referenced by: baseslid 12018 ressval2 12022 ressid 12023 1strbas 12061 2strbasg 12063 2stropg 12064 2strbas1g 12066 rngbaseg 12078 srngbased 12085 lmodbased 12096 ipsbased 12104 topgrpbasd 12114 |
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