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Theorem isstruct2 13406
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 13405 . . 3  |-  Rel Struct
2 brrelex12 4856 . . 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 3454 . . . . 5  |-  F  C_  ( F  u.  { (/) } )
5 undif1 3647 . . . . 5  |-  ( ( F  \  { (/) } )  u.  { (/) } )  =  ( F  u.  { (/) } )
64, 5sseqtr4i 3325 . . . 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 5423 . . . . . . 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 3506 . . . . . . . . . . . 12  |-  (  <_  i^i  ( NN  X.  NN ) )  C_  ( NN  X.  NN )
1110sseli 3288 . . . . . . . . . . 11  |-  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  ->  X  e.  ( NN  X.  NN ) )
12 1st2nd2 6326 . . . . . . . . . . 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 5673 . . . . . . . 8  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  ( ... `  X )  =  ( ... `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
16 df-ov 6024 . . . . . . . . 9  |-  ( ( 1st `  X ) ... ( 2nd `  X
) )  =  ( ... `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. )
17 fzfi 11239 . . . . . . . . 9  |-  ( ( 1st `  X ) ... ( 2nd `  X
) )  e.  Fin
1816, 17eqeltrri 2459 . . . . . . . 8  |-  ( ... `  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )  e.  Fin
1915, 18syl6eqel 2476 . . . . . . 7  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  ( ... `  X )  e. 
Fin )
20 difss 3418 . . . . . . . . 9  |-  ( F 
\  { (/) } ) 
C_  F
21 dmss 5010 . . . . . . . . 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 3303 . . . . . . 7  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  dom  ( F  \  { (/) } )  C_  ( ... `  X ) )
25 ssfi 7266 . . . . . . 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 7321 . . . . . 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 4328 . . . . 5  |-  { (/) }  e.  _V
30 unexg 4651 . . . . 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 4291 . . . 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 2908 . . . 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 2454 . . . 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 3408 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( f  \  { (/)
} )  =  ( F  \  { (/) } ) )
4140funeqd 5416 . . . 4  |-  ( ( f  =  F  /\  x  =  X )  ->  ( Fun  ( f 
\  { (/) } )  <->  Fun  ( F  \  { (/)
} ) ) )
4239dmeqd 5013 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  dom  f  =  dom  F )
4337fveq2d 5673 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( ... `  x
)  =  ( ... `  X ) )
4442, 43sseq12d 3321 . . . 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 13399 . . 3  |- Struct  =  { <. f ,  x >.  |  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f  \  { (/)
} )  /\  dom  f  C_  ( ... `  x
) ) }
4745, 46brabga 4411 . 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 1717   _Vcvv 2900    \ cdif 3261    u. cun 3262    i^i cin 3263    C_ wss 3264   (/)c0 3572   {csn 3758   <.cop 3761   class class class wbr 4154    X. cxp 4817   dom cdm 4819   Rel wrel 4824   Fun wfun 5389    Fn wfn 5390   ` cfv 5395  (class class class)co 6021   1stc1st 6287   2ndc2nd 6288   Fincfn 7046    <_ cle 9055   NNcn 9933   ...cfz 10976   Struct cstr 13393
This theorem is referenced by:  isstruct  13407  structcnvcnv  13408  structfun  13409
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-struct 13399
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