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Theorem isstruct2 13433
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 13432 . . 3  |-  Rel Struct
2 brrelex12 4874 . . 3  |-  ( ( Rel Struct  /\  F Struct  X )  ->  ( F  e.  _V  /\  X  e.  _V )
)
31, 2mpan 652 . 2  |-  ( F Struct  X  ->  ( F  e. 
_V  /\  X  e.  _V ) )
4 ssun1 3470 . . . . 5  |-  F  C_  ( F  u.  { (/) } )
5 undif1 3663 . . . . 5  |-  ( ( F  \  { (/) } )  u.  { (/) } )  =  ( F  u.  { (/) } )
64, 5sseqtr4i 3341 . . . 4  |-  F  C_  ( ( F  \  { (/) } )  u. 
{ (/) } )
7 simp2 958 . . . . . . 7  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  Fun  ( F  \  { (/) } ) )
8 funfn 5441 . . . . . . 7  |-  ( Fun  ( F  \  { (/)
} )  <->  ( F  \  { (/) } )  Fn 
dom  ( F  \  { (/) } ) )
97, 8sylib 189 . . . . . 6  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  ( F  \  { (/) } )  Fn  dom  ( F 
\  { (/) } ) )
10 inss2 3522 . . . . . . . . . . . 12  |-  (  <_  i^i  ( NN  X.  NN ) )  C_  ( NN  X.  NN )
1110sseli 3304 . . . . . . . . . . 11  |-  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  ->  X  e.  ( NN  X.  NN ) )
12 1st2nd2 6345 . . . . . . . . . . 11  |-  ( X  e.  ( NN  X.  NN )  ->  X  = 
<. ( 1st `  X
) ,  ( 2nd `  X ) >. )
1311, 12syl 16 . . . . . . . . . 10  |-  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  ->  X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
14133ad2ant1 978 . . . . . . . . 9  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
1514fveq2d 5691 . . . . . . . 8  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  ( ... `  X )  =  ( ... `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
16 df-ov 6043 . . . . . . . . 9  |-  ( ( 1st `  X ) ... ( 2nd `  X
) )  =  ( ... `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. )
17 fzfi 11266 . . . . . . . . 9  |-  ( ( 1st `  X ) ... ( 2nd `  X
) )  e.  Fin
1816, 17eqeltrri 2475 . . . . . . . 8  |-  ( ... `  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )  e.  Fin
1915, 18syl6eqel 2492 . . . . . . 7  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  ( ... `  X )  e. 
Fin )
20 difss 3434 . . . . . . . . 9  |-  ( F 
\  { (/) } ) 
C_  F
21 dmss 5028 . . . . . . . . 9  |-  ( ( F  \  { (/) } )  C_  F  ->  dom  ( F  \  { (/)
} )  C_  dom  F )
2220, 21ax-mp 8 . . . . . . . 8  |-  dom  ( F  \  { (/) } ) 
C_  dom  F
23 simp3 959 . . . . . . . 8  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  dom  F 
C_  ( ... `  X
) )
2422, 23syl5ss 3319 . . . . . . 7  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  dom  ( F  \  { (/) } )  C_  ( ... `  X ) )
25 ssfi 7288 . . . . . . 7  |-  ( ( ( ... `  X
)  e.  Fin  /\  dom  ( F  \  { (/)
} )  C_  ( ... `  X ) )  ->  dom  ( F  \  { (/) } )  e. 
Fin )
2619, 24, 25syl2anc 643 . . . . . 6  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  dom  ( F  \  { (/) } )  e.  Fin )
27 fnfi 7343 . . . . . 6  |-  ( ( ( F  \  { (/)
} )  Fn  dom  ( F  \  { (/) } )  /\  dom  ( F  \  { (/) } )  e.  Fin )  -> 
( F  \  { (/)
} )  e.  Fin )
289, 26, 27syl2anc 643 . . . . 5  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  ( F  \  { (/) } )  e.  Fin )
29 p0ex 4346 . . . . 5  |-  { (/) }  e.  _V
30 unexg 4669 . . . . 5  |-  ( ( ( F  \  { (/)
} )  e.  Fin  /\ 
{ (/) }  e.  _V )  ->  ( ( F 
\  { (/) } )  u.  { (/) } )  e.  _V )
3128, 29, 30sylancl 644 . . . 4  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  (
( F  \  { (/)
} )  u.  { (/)
} )  e.  _V )
32 ssexg 4309 . . . 4  |-  ( ( F  C_  ( ( F  \  { (/) } )  u.  { (/) } )  /\  ( ( F 
\  { (/) } )  u.  { (/) } )  e.  _V )  ->  F  e.  _V )
336, 31, 32sylancr 645 . . 3  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  F  e.  _V )
34 elex 2924 . . . 4  |-  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  ->  X  e.  _V )
35343ad2ant1 978 . . 3  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  X  e.  _V )
3633, 35jca 519 . 2  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  ( F  e.  _V  /\  X  e.  _V ) )
37 simpr 448 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  x  =  X )
3837eleq1d 2470 . . . 4  |-  ( ( f  =  F  /\  x  =  X )  ->  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  <->  X  e.  (  <_  i^i  ( NN  X.  NN ) ) ) )
39 simpl 444 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  f  =  F )
4039difeq1d 3424 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( f  \  { (/)
} )  =  ( F  \  { (/) } ) )
4140funeqd 5434 . . . 4  |-  ( ( f  =  F  /\  x  =  X )  ->  ( Fun  ( f 
\  { (/) } )  <->  Fun  ( F  \  { (/)
} ) ) )
4239dmeqd 5031 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  dom  f  =  dom  F )
4337fveq2d 5691 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( ... `  x
)  =  ( ... `  X ) )
4442, 43sseq12d 3337 . . . 4  |-  ( ( f  =  F  /\  x  =  X )  ->  ( dom  f  C_  ( ... `  x )  <->  dom  F  C_  ( ... `  X ) ) )
4538, 41, 443anbi123d 1254 . . 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 13426 . . 3  |- Struct  =  { <. f ,  x >.  |  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f  \  { (/)
} )  /\  dom  f  C_  ( ... `  x
) ) }
4745, 46brabga 4429 . 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 343 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774   <.cop 3777   class class class wbr 4172    X. cxp 4835   dom cdm 4837   Rel wrel 4842   Fun wfun 5407    Fn wfn 5408   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   Fincfn 7068    <_ cle 9077   NNcn 9956   ...cfz 10999   Struct cstr 13420
This theorem is referenced by:  isstruct  13434  structcnvcnv  13435  structfun  13436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426
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