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Theorem isstruct2 13470
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 13469 . . 3  |-  Rel Struct
2 brrelex12 4907 . . 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 3502 . . . . 5  |-  F  C_  ( F  u.  { (/) } )
5 undif1 3695 . . . . 5  |-  ( ( F  \  { (/) } )  u.  { (/) } )  =  ( F  u.  { (/) } )
64, 5sseqtr4i 3373 . . . 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 5474 . . . . . . 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 3554 . . . . . . . . . . . 12  |-  (  <_  i^i  ( NN  X.  NN ) )  C_  ( NN  X.  NN )
1110sseli 3336 . . . . . . . . . . 11  |-  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  ->  X  e.  ( NN  X.  NN ) )
12 1st2nd2 6378 . . . . . . . . . . 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 5724 . . . . . . . 8  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  ( ... `  X )  =  ( ... `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
16 df-ov 6076 . . . . . . . . 9  |-  ( ( 1st `  X ) ... ( 2nd `  X
) )  =  ( ... `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. )
17 fzfi 11303 . . . . . . . . 9  |-  ( ( 1st `  X ) ... ( 2nd `  X
) )  e.  Fin
1816, 17eqeltrri 2506 . . . . . . . 8  |-  ( ... `  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )  e.  Fin
1915, 18syl6eqel 2523 . . . . . . 7  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  ( ... `  X )  e. 
Fin )
20 difss 3466 . . . . . . . . 9  |-  ( F 
\  { (/) } ) 
C_  F
21 dmss 5061 . . . . . . . . 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 3351 . . . . . . 7  |-  ( ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X
) )  ->  dom  ( F  \  { (/) } )  C_  ( ... `  X ) )
25 ssfi 7321 . . . . . . 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 7376 . . . . . 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 4378 . . . . 5  |-  { (/) }  e.  _V
30 unexg 4702 . . . . 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 4341 . . . 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 2956 . . . 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 2501 . . . 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 3456 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( f  \  { (/)
} )  =  ( F  \  { (/) } ) )
4140funeqd 5467 . . . 4  |-  ( ( f  =  F  /\  x  =  X )  ->  ( Fun  ( f 
\  { (/) } )  <->  Fun  ( F  \  { (/)
} ) ) )
4239dmeqd 5064 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  dom  f  =  dom  F )
4337fveq2d 5724 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( ... `  x
)  =  ( ... `  X ) )
4442, 43sseq12d 3369 . . . 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 13463 . . 3  |- Struct  =  { <. f ,  x >.  |  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f  \  { (/)
} )  /\  dom  f  C_  ( ... `  x
) ) }
4745, 46brabga 4461 . 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 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806   <.cop 3809   class class class wbr 4204    X. cxp 4868   dom cdm 4870   Rel wrel 4875   Fun wfun 5440    Fn wfn 5441   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   Fincfn 7101    <_ cle 9113   NNcn 9992   ...cfz 11035   Struct cstr 13457
This theorem is referenced by:  isstruct  13471  structcnvcnv  13472  structfun  13473
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463
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