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Theorem strndxid 12024
Description: The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.)
Hypotheses
Ref Expression
strndxid.s  |-  ( ph  ->  S  e.  V )
strndxid.e  |-  E  = Slot 
N
strndxid.n  |-  N  e.  NN
Assertion
Ref Expression
strndxid  |-  ( ph  ->  ( S `  ( E `  ndx ) )  =  ( E `  S ) )

Proof of Theorem strndxid
StepHypRef Expression
1 strndxid.e . . . 4  |-  E  = Slot 
N
2 strndxid.n . . . 4  |-  N  e.  NN
31, 2ndxid 12020 . . 3  |-  E  = Slot  ( E `  ndx )
4 strndxid.s . . 3  |-  ( ph  ->  S  e.  V )
51, 2ndxarg 12019 . . . . 5  |-  ( E `
 ndx )  =  N
65, 2eqeltri 2213 . . . 4  |-  ( E `
 ndx )  e.  NN
76a1i 9 . . 3  |-  ( ph  ->  ( E `  ndx )  e.  NN )
83, 4, 7strnfvnd 12016 . 2  |-  ( ph  ->  ( E `  S
)  =  ( S `
 ( E `  ndx ) ) )
98eqcomd 2146 1  |-  ( ph  ->  ( S `  ( E `  ndx ) )  =  ( E `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332    e. wcel 1481   ` cfv 5130   NNcn 8743   ndxcnx 11993  Slot cslot 11995
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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-cnex 7734  ax-resscn 7735  ax-1re 7737  ax-addrcl 7740
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-iota 5095  df-fun 5132  df-fv 5138  df-inn 8744  df-ndx 11999  df-slot 12000
This theorem is referenced by: (None)
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