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Theorem isstructr 12477
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 4694 . . . 4  |-  ( M (  <_  i^i  ( NN  X.  NN ) ) N  <->  ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N ) )
2 df-br 4005 . . . 4  |-  ( M (  <_  i^i  ( NN  X.  NN ) ) N  <->  <. M ,  N >.  e.  (  <_  i^i  ( NN  X.  NN ) ) )
31, 2sylbb1 137 . . 3  |-  ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  ->  <. M ,  N >.  e.  (  <_  i^i  ( NN  X.  NN ) ) )
43adantr 276 . 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 1003 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  /\  ( Fun  ( F  \  { (/) } )  /\  F  e.  V  /\  dom  F  C_  ( M ... N ) ) )  ->  Fun  ( F  \  { (/) } ) )
6 simpr2 1004 . 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 5878 . . . . . 6  |-  ( M ... N )  =  ( ... `  <. M ,  N >. )
87sseq2i 3183 . . . . 5  |-  ( dom 
F  C_  ( M ... N )  <->  dom  F  C_  ( ... `  <. M ,  N >. ) )
98biimpi 120 . . . 4  |-  ( dom 
F  C_  ( M ... N )  ->  dom  F 
C_  ( ... `  <. M ,  N >. )
)
1093ad2ant3 1020 . . 3  |-  ( ( Fun  ( F  \  { (/) } )  /\  F  e.  V  /\  dom  F  C_  ( M ... N ) )  ->  dom  F  C_  ( ... ` 
<. M ,  N >. ) )
1110adantl 277 . 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 12473 . 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 1239 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 104    /\ w3a 978    e. wcel 2148    \ cdif 3127    i^i cin 3129    C_ wss 3130   (/)c0 3423   {csn 3593   <.cop 3596   class class class wbr 4004    X. cxp 4625   dom cdm 4627   Fun wfun 5211   ` cfv 5217  (class class class)co 5875    <_ cle 7993   NNcn 8919   ...cfz 10008   Struct cstr 12458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-ov 5878  df-struct 12464
This theorem is referenced by:  strleund  12562  strleun  12563  strext  12564  strle1g  12565
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