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Mirrors > Home > ILE Home > Th. List > opelstrsl | GIF version |
Description: The slot of a structure which contains an ordered pair for that slot. (Contributed by Jim Kingdon, 5-Feb-2023.) |
Ref | Expression |
---|---|
opelstrsl.e | ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) |
opelstrsl.s | ⊢ (𝜑 → 𝑆 Struct 𝑋) |
opelstrsl.v | ⊢ (𝜑 → 𝑉 ∈ 𝑌) |
opelstrsl.el | ⊢ (𝜑 → 〈(𝐸‘ndx), 𝑉〉 ∈ 𝑆) |
Ref | Expression |
---|---|
opelstrsl | ⊢ (𝜑 → 𝑉 = (𝐸‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelstrsl.e | . 2 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) | |
2 | opelstrsl.s | . . 3 ⊢ (𝜑 → 𝑆 Struct 𝑋) | |
3 | structex 11981 | . . 3 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (𝜑 → 𝑆 ∈ V) |
5 | structfung 11986 | . . 3 ⊢ (𝑆 Struct 𝑋 → Fun ◡◡𝑆) | |
6 | 2, 5 | syl 14 | . 2 ⊢ (𝜑 → Fun ◡◡𝑆) |
7 | opelstrsl.el | . 2 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝑉〉 ∈ 𝑆) | |
8 | opelstrsl.v | . 2 ⊢ (𝜑 → 𝑉 ∈ 𝑌) | |
9 | 1, 4, 6, 7, 8 | strslfv2d 12011 | 1 ⊢ (𝜑 → 𝑉 = (𝐸‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 Vcvv 2686 〈cop 3530 class class class wbr 3929 ◡ccnv 4538 Fun wfun 5117 ‘cfv 5123 ℕcn 8727 Struct cstr 11965 ndxcnx 11966 Slot cslot 11968 |
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 |
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-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-struct 11971 df-slot 11973 |
This theorem is referenced by: opelstrbas 12066 2strop1g 12074 rngplusgg 12086 rngmulrg 12087 srngplusgd 12093 srngmulrd 12094 srnginvld 12095 lmodplusgd 12104 lmodscad 12105 lmodvscad 12106 ipsaddgd 12112 ipsmulrd 12113 ipsscad 12114 ipsvscad 12115 ipsipd 12116 topgrpplusgd 12122 topgrptsetd 12123 |
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