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Theorem isstruct2 13157
Description: The property of being a structure with components in  ( 1st `  X
) ... ( 2nd `  X
). (Contributed by Mario Carneiro, 29-Aug-2015.)
Assertion
Ref Expression
isstruct2  |-  ( F Struct  X 
<->  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/)
} )  /\  dom  F 
C_  ( ... `  X
) ) )

Proof of Theorem isstruct2
Dummy variables  x  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brstruct 13156 . . 3  |-  Rel Struct
2 brrelex12 4726 . . 3  |-  ( ( Rel Struct  /\  F Struct  X )  ->  ( F  e.  _V  /\  X  e.  _V )
)
31, 2mpan 651 . 2  |-  ( F Struct  X  ->  ( F  e. 
_V  /\  X  e.  _V ) )
4 ssun1 3338 . . . . 5  |-  F  C_  ( F  u.  { (/) } )
5 undif1 3529 . . . . 5  |-  ( ( F  \  { (/) } )  u.  { (/) } )  =  ( F  u.  { (/) } )
64, 5sseqtr4i 3211 . . . 4  |-  F  C_  ( ( F  \  { (/) } )  u. 
{ (/) } )
7 simp2 956 . . . . . . 7  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  Fun  ( F  \  { (/) } ) )
8 funfn 5283 . . . . . . 7  |-  ( Fun  ( F  \  { (/)
} )  <->  ( F  \  { (/) } )  Fn 
dom  ( F  \  { (/) } ) )
97, 8sylib 188 . . . . . 6  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  ( F  \  { (/) } )  Fn  dom  ( F 
\  { (/) } ) )
10 inss2 3390 . . . . . . . . . . . 12  |-  (  <_  i^i  ( NN  X.  NN ) )  C_  ( NN  X.  NN )
1110sseli 3176 . . . . . . . . . . 11  |-  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  ->  X  e.  ( NN  X.  NN ) )
12 1st2nd2 6159 . . . . . . . . . . 11  |-  ( X  e.  ( NN  X.  NN )  ->  X  = 
<. ( 1st `  X
) ,  ( 2nd `  X ) >. )
1311, 12syl 15 . . . . . . . . . 10  |-  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  ->  X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
14133ad2ant1 976 . . . . . . . . 9  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
1514fveq2d 5529 . . . . . . . 8  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  ( ... `  X )  =  ( ... `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
16 df-ov 5861 . . . . . . . . 9  |-  ( ( 1st `  X ) ... ( 2nd `  X
) )  =  ( ... `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. )
17 fzfi 11034 . . . . . . . . 9  |-  ( ( 1st `  X ) ... ( 2nd `  X
) )  e.  Fin
1816, 17eqeltrri 2354 . . . . . . . 8  |-  ( ... `  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )  e.  Fin
1915, 18syl6eqel 2371 . . . . . . 7  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  ( ... `  X )  e. 
Fin )
20 difss 3303 . . . . . . . . 9  |-  ( F 
\  { (/) } ) 
C_  F
21 dmss 4878 . . . . . . . . 9  |-  ( ( F  \  { (/) } )  C_  F  ->  dom  ( F  \  { (/)
} )  C_  dom  F )
2220, 21ax-mp 8 . . . . . . . 8  |-  dom  ( F  \  { (/) } ) 
C_  dom  F
23 simp3 957 . . . . . . . 8  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  dom  F 
C_  ( ... `  X
) )
2422, 23syl5ss 3190 . . . . . . 7  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  dom  ( F  \  { (/) } )  C_  ( ... `  X ) )
25 ssfi 7083 . . . . . . 7  |-  ( ( ( ... `  X
)  e.  Fin  /\  dom  ( F  \  { (/)
} )  C_  ( ... `  X ) )  ->  dom  ( F  \  { (/) } )  e. 
Fin )
2619, 24, 25syl2anc 642 . . . . . 6  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  dom  ( F  \  { (/) } )  e.  Fin )
27 fnfi 7134 . . . . . 6  |-  ( ( ( F  \  { (/)
} )  Fn  dom  ( F  \  { (/) } )  /\  dom  ( F  \  { (/) } )  e.  Fin )  -> 
( F  \  { (/)
} )  e.  Fin )
289, 26, 27syl2anc 642 . . . . 5  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  ( F  \  { (/) } )  e.  Fin )
29 p0ex 4197 . . . . 5  |-  { (/) }  e.  _V
30 unexg 4521 . . . . 5  |-  ( ( ( F  \  { (/)
} )  e.  Fin  /\ 
{ (/) }  e.  _V )  ->  ( ( F 
\  { (/) } )  u.  { (/) } )  e.  _V )
3128, 29, 30sylancl 643 . . . 4  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  (
( F  \  { (/)
} )  u.  { (/)
} )  e.  _V )
32 ssexg 4160 . . . 4  |-  ( ( F  C_  ( ( F  \  { (/) } )  u.  { (/) } )  /\  ( ( F 
\  { (/) } )  u.  { (/) } )  e.  _V )  ->  F  e.  _V )
336, 31, 32sylancr 644 . . 3  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  F  e.  _V )
34 elex 2796 . . . 4  |-  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  ->  X  e.  _V )
35343ad2ant1 976 . . 3  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  X  e.  _V )
3633, 35jca 518 . 2  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  ( F  e.  _V  /\  X  e.  _V ) )
37 simpr 447 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  x  =  X )
3837eleq1d 2349 . . . 4  |-  ( ( f  =  F  /\  x  =  X )  ->  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  <->  X  e.  (  <_  i^i  ( NN  X.  NN ) ) ) )
39 simpl 443 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  f  =  F )
4039difeq1d 3293 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( f  \  { (/)
} )  =  ( F  \  { (/) } ) )
4140funeqd 5276 . . . 4  |-  ( ( f  =  F  /\  x  =  X )  ->  ( Fun  ( f 
\  { (/) } )  <->  Fun  ( F  \  { (/)
} ) ) )
4239dmeqd 4881 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  dom  f  =  dom  F )
4337fveq2d 5529 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( ... `  x
)  =  ( ... `  X ) )
4442, 43sseq12d 3207 . . . 4  |-  ( ( f  =  F  /\  x  =  X )  ->  ( dom  f  C_  ( ... `  x )  <->  dom  F  C_  ( ... `  X ) ) )
4538, 41, 443anbi123d 1252 . . 3  |-  ( ( 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 ) ) ) )
46 df-struct 13150 . . 3  |- Struct  =  { <. f ,  x >.  |  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f  \  { (/)
} )  /\  dom  f  C_  ( ... `  x
) ) }
4745, 46brabga 4279 . 2  |-  ( ( F  e.  _V  /\  X  e.  _V )  ->  ( F Struct  X  <->  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F 
\  { (/) } )  /\  dom  F  C_  ( ... `  X ) ) ) )
483, 36, 47pm5.21nii 342 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:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   class class class wbr 4023    X. cxp 4687   dom cdm 4689   Rel wrel 4694   Fun wfun 5249    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   Fincfn 6863    <_ cle 8868   NNcn 9746   ...cfz 10782   Struct cstr 13144
This theorem is referenced by:  isstruct  13158  structcnvcnv  13159  structfun  13160
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150
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