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Theorem opelstrsl 12565
Description: The slot of a structure which contains an ordered pair for that slot. (Contributed by Jim Kingdon, 5-Feb-2023.)
Hypotheses
Ref Expression
opelstrsl.e  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
opelstrsl.s  |-  ( ph  ->  S Struct  X )
opelstrsl.v  |-  ( ph  ->  V  e.  Y )
opelstrsl.el  |-  ( ph  -> 
<. ( E `  ndx ) ,  V >.  e.  S )
Assertion
Ref Expression
opelstrsl  |-  ( ph  ->  V  =  ( E `
 S ) )

Proof of Theorem opelstrsl
StepHypRef Expression
1 opelstrsl.e . 2  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
2 opelstrsl.s . . 3  |-  ( ph  ->  S Struct  X )
3 structex 12466 . . 3  |-  ( S Struct  X  ->  S  e.  _V )
42, 3syl 14 . 2  |-  ( ph  ->  S  e.  _V )
5 structfung 12471 . . 3  |-  ( S Struct  X  ->  Fun  `' `' S )
62, 5syl 14 . 2  |-  ( ph  ->  Fun  `' `' S
)
7 opelstrsl.el . 2  |-  ( ph  -> 
<. ( E `  ndx ) ,  V >.  e.  S )
8 opelstrsl.v . 2  |-  ( ph  ->  V  e.  Y )
91, 4, 6, 7, 8strslfv2d 12497 1  |-  ( ph  ->  V  =  ( E `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   _Vcvv 2737   <.cop 3595   class class class wbr 4002   `'ccnv 4624   Fun wfun 5209   ` cfv 5215   NNcn 8915   Struct cstr 12450   ndxcnx 12451  Slot cslot 12453
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 4120  ax-pow 4173  ax-pr 4208  ax-un 4432
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 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5177  df-fun 5217  df-fv 5223  df-struct 12456  df-slot 12458
This theorem is referenced by:  opelstrbas  12566  2strop1g  12574  rngplusgg  12587  rngmulrg  12588  srngplusgd  12598  srngmulrd  12599  srnginvld  12600  lmodplusgd  12616  lmodscad  12617  lmodvscad  12618  ipsaddgd  12628  ipsmulrd  12629  ipsscad  12630  ipsvscad  12631  ipsipd  12632  topgrpplusgd  12645  topgrptsetd  12646
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