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Theorem isstructr 12538
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 4714 . . . 4  |-  ( M (  <_  i^i  ( NN  X.  NN ) ) N  <->  ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N ) )
2 df-br 4022 . . . 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 1005 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  /\  ( Fun  ( F  \  { (/) } )  /\  F  e.  V  /\  dom  F  C_  ( M ... N ) ) )  ->  Fun  ( F  \  { (/) } ) )
6 simpr2 1006 . 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 5903 . . . . . 6  |-  ( M ... N )  =  ( ... `  <. M ,  N >. )
87sseq2i 3197 . . . . 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 1022 . . 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 12534 . 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 1250 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 980    e. wcel 2160    \ cdif 3141    i^i cin 3143    C_ wss 3144   (/)c0 3437   {csn 3610   <.cop 3613   class class class wbr 4021    X. cxp 4645   dom cdm 4647   Fun wfun 5232   ` cfv 5238  (class class class)co 5900    <_ cle 8028   NNcn 8954   ...cfz 10044   Struct cstr 12519
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-iota 5199  df-fun 5240  df-fv 5246  df-ov 5903  df-struct 12525
This theorem is referenced by:  strleund  12626  strleun  12627  strext  12628  strle1g  12629
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