MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isstruct2 Unicode version

Theorem isstruct2 13173
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 13172 . . 3  |-  Rel Struct
2 brrelex12 4742 . . 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 3351 . . . . 5  |-  F  C_  ( F  u.  { (/) } )
5 undif1 3542 . . . . 5  |-  ( ( F  \  { (/) } )  u.  { (/) } )  =  ( F  u.  { (/) } )
64, 5sseqtr4i 3224 . . . 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 5299 . . . . . . 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 3403 . . . . . . . . . . . 12  |-  (  <_  i^i  ( NN  X.  NN ) )  C_  ( NN  X.  NN )
1110sseli 3189 . . . . . . . . . . 11  |-  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  ->  X  e.  ( NN  X.  NN ) )
12 1st2nd2 6175 . . . . . . . . . . 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 5545 . . . . . . . 8  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  ( ... `  X )  =  ( ... `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
16 df-ov 5877 . . . . . . . . 9  |-  ( ( 1st `  X ) ... ( 2nd `  X
) )  =  ( ... `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. )
17 fzfi 11050 . . . . . . . . 9  |-  ( ( 1st `  X ) ... ( 2nd `  X
) )  e.  Fin
1816, 17eqeltrri 2367 . . . . . . . 8  |-  ( ... `  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )  e.  Fin
1915, 18syl6eqel 2384 . . . . . . 7  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  ( ... `  X )  e. 
Fin )
20 difss 3316 . . . . . . . . 9  |-  ( F 
\  { (/) } ) 
C_  F
21 dmss 4894 . . . . . . . . 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 3203 . . . . . . 7  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  dom  ( F  \  { (/) } )  C_  ( ... `  X ) )
25 ssfi 7099 . . . . . . 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 7150 . . . . . 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 4213 . . . . 5  |-  { (/) }  e.  _V
30 unexg 4537 . . . . 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 4176 . . . 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 2809 . . . 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 2362 . . . 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 3306 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( f  \  { (/)
} )  =  ( F  \  { (/) } ) )
4140funeqd 5292 . . . 4  |-  ( ( f  =  F  /\  x  =  X )  ->  ( Fun  ( f 
\  { (/) } )  <->  Fun  ( F  \  { (/)
} ) ) )
4239dmeqd 4897 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  dom  f  =  dom  F )
4337fveq2d 5545 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( ... `  x
)  =  ( ... `  X ) )
4442, 43sseq12d 3220 . . . 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 13166 . . 3  |- Struct  =  { <. f ,  x >.  |  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f  \  { (/)
} )  /\  dom  f  C_  ( ... `  x
) ) }
4745, 46brabga 4295 . 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 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   <.cop 3656   class class class wbr 4039    X. cxp 4703   dom cdm 4705   Rel wrel 4710   Fun wfun 5265    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   Fincfn 6879    <_ cle 8884   NNcn 9762   ...cfz 10798   Struct cstr 13160
This theorem is referenced by:  isstruct  13174  structcnvcnv  13175  structfun  13176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166
  Copyright terms: Public domain W3C validator