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Theorem isstructr 11756
Description: The property of being a structure with components in  M ... N. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
Assertion
Ref Expression
isstructr  |-  ( ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  /\  ( Fun  ( F  \  { (/) } )  /\  F  e.  V  /\  dom  F  C_  ( M ... N ) ) )  ->  F Struct  <. M ,  N >. )

Proof of Theorem isstructr
StepHypRef Expression
1 brinxp2 4544 . . . 4  |-  ( M (  <_  i^i  ( NN  X.  NN ) ) N  <->  ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N ) )
2 df-br 3876 . . . 4  |-  ( M (  <_  i^i  ( NN  X.  NN ) ) N  <->  <. M ,  N >.  e.  (  <_  i^i  ( NN  X.  NN ) ) )
31, 2sylbb1 136 . . 3  |-  ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  ->  <. M ,  N >.  e.  (  <_  i^i  ( NN  X.  NN ) ) )
43adantr 272 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  /\  ( Fun  ( F  \  { (/) } )  /\  F  e.  V  /\  dom  F  C_  ( M ... N ) ) )  ->  <. M ,  N >.  e.  (  <_  i^i  ( NN  X.  NN ) ) )
5 simpr1 955 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  /\  ( Fun  ( F  \  { (/) } )  /\  F  e.  V  /\  dom  F  C_  ( M ... N ) ) )  ->  Fun  ( F  \  { (/) } ) )
6 simpr2 956 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  /\  ( Fun  ( F  \  { (/) } )  /\  F  e.  V  /\  dom  F  C_  ( M ... N ) ) )  ->  F  e.  V
)
7 df-ov 5709 . . . . . 6  |-  ( M ... N )  =  ( ... `  <. M ,  N >. )
87sseq2i 3074 . . . . 5  |-  ( dom 
F  C_  ( M ... N )  <->  dom  F  C_  ( ... `  <. M ,  N >. ) )
98biimpi 119 . . . 4  |-  ( dom 
F  C_  ( M ... N )  ->  dom  F 
C_  ( ... `  <. M ,  N >. )
)
1093ad2ant3 972 . . 3  |-  ( ( Fun  ( F  \  { (/) } )  /\  F  e.  V  /\  dom  F  C_  ( M ... N ) )  ->  dom  F  C_  ( ... ` 
<. M ,  N >. ) )
1110adantl 273 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  /\  ( Fun  ( F  \  { (/) } )  /\  F  e.  V  /\  dom  F  C_  ( M ... N ) ) )  ->  dom  F  C_  ( ... `  <. M ,  N >. ) )
12 isstruct2r 11752 . 2  |-  ( ( ( <. M ,  N >.  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } ) )  /\  ( F  e.  V  /\  dom  F  C_  ( ... ` 
<. M ,  N >. ) ) )  ->  F Struct  <. M ,  N >. )
134, 5, 6, 11, 12syl22anc 1185 1  |-  ( ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  /\  ( Fun  ( F  \  { (/) } )  /\  F  e.  V  /\  dom  F  C_  ( M ... N ) ) )  ->  F Struct  <. M ,  N >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 930    e. wcel 1448    \ cdif 3018    i^i cin 3020    C_ wss 3021   (/)c0 3310   {csn 3474   <.cop 3477   class class class wbr 3875    X. cxp 4475   dom cdm 4477   Fun wfun 5053   ` cfv 5059  (class class class)co 5706    <_ cle 7673   NNcn 8578   ...cfz 9631   Struct cstr 11737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-ov 5709  df-struct 11743
This theorem is referenced by:  strleund  11829  strleun  11830  strle1g  11831
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