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Theorem isstruct2im 11979
Description: The property of being a structure with components in  ( 1st `  X
) ... ( 2nd `  X
). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
Assertion
Ref Expression
isstruct2im  |-  ( F Struct  X  ->  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F 
\  { (/) } )  /\  dom  F  C_  ( ... `  X ) ) )

Proof of Theorem isstruct2im
Dummy variables  x  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brstruct 11978 . . . 4  |-  Rel Struct
21brrelex12i 4581 . . 3  |-  ( F Struct  X  ->  ( F  e. 
_V  /\  X  e.  _V ) )
3 simpr 109 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  x  =  X )
43eleq1d 2208 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  <->  X  e.  (  <_  i^i  ( NN  X.  NN ) ) ) )
5 simpl 108 . . . . . . 7  |-  ( ( f  =  F  /\  x  =  X )  ->  f  =  F )
65difeq1d 3193 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  ( f  \  { (/)
} )  =  ( F  \  { (/) } ) )
76funeqd 5145 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( Fun  ( f 
\  { (/) } )  <->  Fun  ( F  \  { (/)
} ) ) )
85dmeqd 4741 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  dom  f  =  dom  F )
93fveq2d 5425 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  ( ... `  x
)  =  ( ... `  X ) )
108, 9sseq12d 3128 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( dom  f  C_  ( ... `  x )  <->  dom  F  C_  ( ... `  X ) ) )
114, 7, 103anbi123d 1290 . . . 4  |-  ( ( f  =  F  /\  x  =  X )  ->  ( ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f 
\  { (/) } )  /\  dom  f  C_  ( ... `  x ) )  <->  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F 
\  { (/) } )  /\  dom  F  C_  ( ... `  X ) ) ) )
12 df-struct 11971 . . . 4  |- Struct  =  { <. f ,  x >.  |  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f  \  { (/)
} )  /\  dom  f  C_  ( ... `  x
) ) }
1311, 12brabga 4186 . . 3  |-  ( ( F  e.  _V  /\  X  e.  _V )  ->  ( F Struct  X  <->  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F 
\  { (/) } )  /\  dom  F  C_  ( ... `  X ) ) ) )
142, 13syl 14 . 2  |-  ( F Struct  X  ->  ( F Struct  X  <->  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) ) ) )
1514ibi 175 1  |-  ( F Struct  X  ->  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F 
\  { (/) } )  /\  dom  F  C_  ( ... `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   _Vcvv 2686    \ cdif 3068    i^i cin 3070    C_ wss 3071   (/)c0 3363   {csn 3527   class class class wbr 3929    X. cxp 4537   dom cdm 4539   Fun wfun 5117   ` cfv 5123    <_ cle 7808   NNcn 8727   ...cfz 9797   Struct cstr 11965
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-struct 11971
This theorem is referenced by:  structn0fun  11982  isstructim  11983
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