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Theorem opelstrsl 12573
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 12474 . . 3  |-  ( S Struct  X  ->  S  e.  _V )
42, 3syl 14 . 2  |-  ( ph  ->  S  e.  _V )
5 structfung 12479 . . 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 12505 1  |-  ( ph  ->  V  =  ( E `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   _Vcvv 2738   <.cop 3596   class class class wbr 4004   `'ccnv 4626   Fun wfun 5211   ` cfv 5217   NNcn 8919   Struct cstr 12458   ndxcnx 12459  Slot cslot 12461
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 4122  ax-pow 4175  ax-pr 4210  ax-un 4434
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 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-iota 5179  df-fun 5219  df-fv 5225  df-struct 12464  df-slot 12466
This theorem is referenced by:  opelstrbas  12574  2strop1g  12582  rngplusgg  12595  rngmulrg  12596  srngplusgd  12606  srngmulrd  12607  srnginvld  12608  lmodplusgd  12624  lmodscad  12625  lmodvscad  12626  ipsaddgd  12636  ipsmulrd  12637  ipsscad  12638  ipsvscad  12639  ipsipd  12640  topgrpplusgd  12653  topgrptsetd  12654
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