![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 | β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) |
opelstrsl.s | β’ (π β π Struct π) |
opelstrsl.v | β’ (π β π β π) |
opelstrsl.el | β’ (π β β¨(πΈβndx), πβ© β π) |
Ref | Expression |
---|---|
opelstrsl | β’ (π β π = (πΈβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelstrsl.e | . 2 β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) | |
2 | opelstrsl.s | . . 3 β’ (π β π Struct π) | |
3 | structex 12473 | . . 3 β’ (π Struct π β π β V) | |
4 | 2, 3 | syl 14 | . 2 β’ (π β π β V) |
5 | structfung 12478 | . . 3 β’ (π Struct π β Fun β‘β‘π) | |
6 | 2, 5 | syl 14 | . 2 β’ (π β Fun β‘β‘π) |
7 | opelstrsl.el | . 2 β’ (π β β¨(πΈβndx), πβ© β π) | |
8 | opelstrsl.v | . 2 β’ (π β π β π) | |
9 | 1, 4, 6, 7, 8 | strslfv2d 12504 | 1 β’ (π β π = (πΈβπ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 Vcvv 2737 β¨cop 3595 class class class wbr 4003 β‘ccnv 4625 Fun wfun 5210 βcfv 5216 βcn 8918 Struct cstr 12457 ndxcnx 12458 Slot cslot 12460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fun 5218 df-fv 5224 df-struct 12463 df-slot 12465 |
This theorem is referenced by: opelstrbas 12573 2strop1g 12581 rngplusgg 12594 rngmulrg 12595 srngplusgd 12605 srngmulrd 12606 srnginvld 12607 lmodplusgd 12623 lmodscad 12624 lmodvscad 12625 ipsaddgd 12635 ipsmulrd 12636 ipsscad 12637 ipsvscad 12638 ipsipd 12639 topgrpplusgd 12652 topgrptsetd 12653 |
Copyright terms: Public domain | W3C validator |