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Mirrors > Home > ILE Home > Th. List > opelstrbas | GIF version |
Description: The base set of a structure with a base set. (Contributed by AV, 10-Nov-2021.) |
Ref | Expression |
---|---|
opelstrbas.s | ⊢ (𝜑 → 𝑆 Struct 𝑋) |
opelstrbas.v | ⊢ (𝜑 → 𝑉 ∈ 𝑌) |
opelstrbas.b | ⊢ (𝜑 → 〈(Base‘ndx), 𝑉〉 ∈ 𝑆) |
Ref | Expression |
---|---|
opelstrbas | ⊢ (𝜑 → 𝑉 = (Base‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baseslid 12015 | . 2 ⊢ (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ) | |
2 | opelstrbas.s | . 2 ⊢ (𝜑 → 𝑆 Struct 𝑋) | |
3 | opelstrbas.v | . 2 ⊢ (𝜑 → 𝑉 ∈ 𝑌) | |
4 | opelstrbas.b | . 2 ⊢ (𝜑 → 〈(Base‘ndx), 𝑉〉 ∈ 𝑆) | |
5 | 1, 2, 3, 4 | opelstrsl 12055 | 1 ⊢ (𝜑 → 𝑉 = (Base‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 〈cop 3530 class class class wbr 3929 ‘cfv 5123 Struct cstr 11955 ndxcnx 11956 Basecbs 11959 |
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-in1 603 ax-in2 604 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 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 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-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 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 8721 df-struct 11961 df-ndx 11962 df-slot 11963 df-base 11965 |
This theorem is referenced by: 2strbas1g 12063 rngbaseg 12075 srngbased 12082 lmodbased 12093 ipsbased 12101 topgrpbasd 12111 |
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