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Theorem isstructr 12431
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 4678 . . . 4  |-  ( M (  <_  i^i  ( NN  X.  NN ) ) N  <->  ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N ) )
2 df-br 3990 . . . 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 998 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  /\  ( Fun  ( F  \  { (/) } )  /\  F  e.  V  /\  dom  F  C_  ( M ... N ) ) )  ->  Fun  ( F  \  { (/) } ) )
6 simpr2 999 . 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 5856 . . . . . 6  |-  ( M ... N )  =  ( ... `  <. M ,  N >. )
87sseq2i 3174 . . . . 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 1015 . . 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 12427 . 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 1234 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 973    e. wcel 2141    \ cdif 3118    i^i cin 3120    C_ wss 3121   (/)c0 3414   {csn 3583   <.cop 3586   class class class wbr 3989    X. cxp 4609   dom cdm 4611   Fun wfun 5192   ` cfv 5198  (class class class)co 5853    <_ cle 7955   NNcn 8878   ...cfz 9965   Struct cstr 12412
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-struct 12418
This theorem is referenced by:  strleund  12506  strleun  12507  strle1g  12508
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