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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 12349 | . . 3 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (𝜑 → 𝑆 ∈ V) |
5 | structfung 12354 | . . 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 12379 | 1 ⊢ (𝜑 → 𝑉 = (𝐸‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 Vcvv 2721 〈cop 3573 class class class wbr 3976 ◡ccnv 4597 Fun wfun 5176 ‘cfv 5182 ℕcn 8848 Struct cstr 12333 ndxcnx 12334 Slot cslot 12336 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-iota 5147 df-fun 5184 df-fv 5190 df-struct 12339 df-slot 12341 |
This theorem is referenced by: opelstrbas 12434 2strop1g 12442 rngplusgg 12454 rngmulrg 12455 srngplusgd 12461 srngmulrd 12462 srnginvld 12463 lmodplusgd 12472 lmodscad 12473 lmodvscad 12474 ipsaddgd 12480 ipsmulrd 12481 ipsscad 12482 ipsvscad 12483 ipsipd 12484 topgrpplusgd 12490 topgrptsetd 12491 |
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