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Theorem inftmrel 28332
Description: The infinitesimal relation for a structure  W. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypothesis
Ref Expression
inftm.b  |-  B  =  ( Base `  W
)
Assertion
Ref Expression
inftmrel  |-  ( W  e.  V  ->  (<<< `  W )  C_  ( B  X.  B ) )

Proof of Theorem inftmrel
Dummy variables  x  w  y  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3087 . . 3  |-  ( W  e.  V  ->  W  e.  _V )
2 fveq2 5872 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
3 inftm.b . . . . . . . . 9  |-  B  =  ( Base `  W
)
42, 3syl6eqr 2479 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  B )
54eleq2d 2490 . . . . . . 7  |-  ( w  =  W  ->  (
x  e.  ( Base `  w )  <->  x  e.  B ) )
64eleq2d 2490 . . . . . . 7  |-  ( w  =  W  ->  (
y  e.  ( Base `  w )  <->  y  e.  B ) )
75, 6anbi12d 715 . . . . . 6  |-  ( w  =  W  ->  (
( x  e.  (
Base `  w )  /\  y  e.  ( Base `  w ) )  <-> 
( x  e.  B  /\  y  e.  B
) ) )
8 fveq2 5872 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
9 fveq2 5872 . . . . . . . 8  |-  ( w  =  W  ->  ( lt `  w )  =  ( lt `  W
) )
10 eqidd 2421 . . . . . . . 8  |-  ( w  =  W  ->  x  =  x )
118, 9, 10breq123d 4431 . . . . . . 7  |-  ( w  =  W  ->  (
( 0g `  w
) ( lt `  w ) x  <->  ( 0g `  W ) ( lt
`  W ) x ) )
12 fveq2 5872 . . . . . . . . . 10  |-  ( w  =  W  ->  (.g `  w )  =  (.g `  W ) )
1312oveqd 6313 . . . . . . . . 9  |-  ( w  =  W  ->  (
n (.g `  w ) x )  =  ( n (.g `  W ) x ) )
14 eqidd 2421 . . . . . . . . 9  |-  ( w  =  W  ->  y  =  y )
1513, 9, 14breq123d 4431 . . . . . . . 8  |-  ( w  =  W  ->  (
( n (.g `  w
) x ) ( lt `  w ) y  <->  ( n (.g `  W ) x ) ( lt `  W
) y ) )
1615ralbidv 2862 . . . . . . 7  |-  ( w  =  W  ->  ( A. n  e.  NN  ( n (.g `  w
) x ) ( lt `  w ) y  <->  A. n  e.  NN  ( n (.g `  W
) x ) ( lt `  W ) y ) )
1711, 16anbi12d 715 . . . . . 6  |-  ( w  =  W  ->  (
( ( 0g `  w ) ( lt
`  w ) x  /\  A. n  e.  NN  ( n (.g `  w ) x ) ( lt `  w
) y )  <->  ( ( 0g `  W ) ( lt `  W ) x  /\  A. n  e.  NN  ( n (.g `  W ) x ) ( lt `  W
) y ) ) )
187, 17anbi12d 715 . . . . 5  |-  ( w  =  W  ->  (
( ( x  e.  ( Base `  w
)  /\  y  e.  ( Base `  w )
)  /\  ( ( 0g `  w ) ( lt `  w ) x  /\  A. n  e.  NN  ( n (.g `  w ) x ) ( lt `  w
) y ) )  <-> 
( ( x  e.  B  /\  y  e.  B )  /\  (
( 0g `  W
) ( lt `  W ) x  /\  A. n  e.  NN  (
n (.g `  W ) x ) ( lt `  W ) y ) ) ) )
1918opabbidv 4480 . . . 4  |-  ( w  =  W  ->  { <. x ,  y >.  |  ( ( x  e.  (
Base `  w )  /\  y  e.  ( Base `  w ) )  /\  ( ( 0g
`  w ) ( lt `  w ) x  /\  A. n  e.  NN  ( n (.g `  w ) x ) ( lt `  w
) y ) ) }  =  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  ( ( 0g `  W ) ( lt `  W ) x  /\  A. n  e.  NN  ( n (.g `  W ) x ) ( lt `  W
) y ) ) } )
20 df-inftm 28330 . . . 4  |- <<<  =  (
w  e.  _V  |->  {
<. x ,  y >.  |  ( ( x  e.  ( Base `  w
)  /\  y  e.  ( Base `  w )
)  /\  ( ( 0g `  w ) ( lt `  w ) x  /\  A. n  e.  NN  ( n (.g `  w ) x ) ( lt `  w
) y ) ) } )
21 fvex 5882 . . . . . . 7  |-  ( Base `  W )  e.  _V
223, 21eqeltri 2504 . . . . . 6  |-  B  e. 
_V
2322, 22xpex 6600 . . . . 5  |-  ( B  X.  B )  e. 
_V
24 opabssxp 4920 . . . . 5  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  ( ( 0g `  W ) ( lt `  W ) x  /\  A. n  e.  NN  ( n (.g `  W ) x ) ( lt `  W
) y ) ) }  C_  ( B  X.  B )
2523, 24ssexi 4561 . . . 4  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  ( ( 0g `  W ) ( lt `  W ) x  /\  A. n  e.  NN  ( n (.g `  W ) x ) ( lt `  W
) y ) ) }  e.  _V
2619, 20, 25fvmpt 5955 . . 3  |-  ( W  e.  _V  ->  (<<< `  W )  =  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B )  /\  (
( 0g `  W
) ( lt `  W ) x  /\  A. n  e.  NN  (
n (.g `  W ) x ) ( lt `  W ) y ) ) } )
271, 26syl 17 . 2  |-  ( W  e.  V  ->  (<<< `  W )  =  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B )  /\  (
( 0g `  W
) ( lt `  W ) x  /\  A. n  e.  NN  (
n (.g `  W ) x ) ( lt `  W ) y ) ) } )
2827, 24syl6eqss 3511 1  |-  ( W  e.  V  ->  (<<< `  W )  C_  ( B  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   _Vcvv 3078    C_ wss 3433   class class class wbr 4417   {copab 4474    X. cxp 4843   ` cfv 5592  (class class class)co 6296   NNcn 10598   Basecbs 15073   0gc0g 15290   ltcplt 16130  .gcmg 16616  <<<cinftm 28328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5556  df-fun 5594  df-fv 5600  df-ov 6299  df-inftm 28330
This theorem is referenced by:  isarchi  28334
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