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Theorem isstructr 12004
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 4610 . . . 4  |-  ( M (  <_  i^i  ( NN  X.  NN ) ) N  <->  ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N ) )
2 df-br 3934 . . . 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 274 . 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 988 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  /\  ( Fun  ( F  \  { (/) } )  /\  F  e.  V  /\  dom  F  C_  ( M ... N ) ) )  ->  Fun  ( F  \  { (/) } ) )
6 simpr2 989 . 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 5781 . . . . . 6  |-  ( M ... N )  =  ( ... `  <. M ,  N >. )
87sseq2i 3125 . . . . 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 1005 . . 3  |-  ( ( Fun  ( F  \  { (/) } )  /\  F  e.  V  /\  dom  F  C_  ( M ... N ) )  ->  dom  F  C_  ( ... ` 
<. M ,  N >. ) )
1110adantl 275 . 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 12000 . 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 1218 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 963    e. wcel 1481    \ cdif 3069    i^i cin 3071    C_ wss 3072   (/)c0 3364   {csn 3528   <.cop 3531   class class class wbr 3933    X. cxp 4541   dom cdm 4543   Fun wfun 5121   ` cfv 5127  (class class class)co 5778    <_ cle 7821   NNcn 8740   ...cfz 9817   Struct cstr 11985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2689  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-br 3934  df-opab 3994  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-iota 5092  df-fun 5129  df-fv 5135  df-ov 5781  df-struct 11991
This theorem is referenced by:  strleund  12077  strleun  12078  strle1g  12079
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