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Theorem isstruct2im 12413
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 12412 . . . 4  |-  Rel Struct
21brrelex12i 4651 . . 3  |-  ( F Struct  X  ->  ( F  e. 
_V  /\  X  e.  _V ) )
3 simpr 109 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  x  =  X )
43eleq1d 2239 . . . . 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 3244 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  ( f  \  { (/)
} )  =  ( F  \  { (/) } ) )
76funeqd 5218 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( Fun  ( f 
\  { (/) } )  <->  Fun  ( F  \  { (/)
} ) ) )
85dmeqd 4811 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  dom  f  =  dom  F )
93fveq2d 5498 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  ( ... `  x
)  =  ( ... `  X ) )
108, 9sseq12d 3178 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( dom  f  C_  ( ... `  x )  <->  dom  F  C_  ( ... `  X ) ) )
114, 7, 103anbi123d 1307 . . . 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 12405 . . . 4  |- Struct  =  { <. f ,  x >.  |  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f  \  { (/)
} )  /\  dom  f  C_  ( ... `  x
) ) }
1311, 12brabga 4247 . . 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 973    = wceq 1348    e. wcel 2141   _Vcvv 2730    \ cdif 3118    i^i cin 3120    C_ wss 3121   (/)c0 3414   {csn 3581   class class class wbr 3987    X. cxp 4607   dom cdm 4609   Fun wfun 5190   ` cfv 5196    <_ cle 7942   NNcn 8865   ...cfz 9952   Struct cstr 12399
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-struct 12405
This theorem is referenced by:  structn0fun  12416  isstructim  12417
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