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Theorem inftmrel 27483
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 3122 . . 3  |-  ( W  e.  V  ->  W  e.  _V )
2 fveq2 5866 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
3 inftm.b . . . . . . . . 9  |-  B  =  ( Base `  W
)
42, 3syl6eqr 2526 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  B )
54eleq2d 2537 . . . . . . 7  |-  ( w  =  W  ->  (
x  e.  ( Base `  w )  <->  x  e.  B ) )
64eleq2d 2537 . . . . . . 7  |-  ( w  =  W  ->  (
y  e.  ( Base `  w )  <->  y  e.  B ) )
75, 6anbi12d 710 . . . . . 6  |-  ( w  =  W  ->  (
( x  e.  (
Base `  w )  /\  y  e.  ( Base `  w ) )  <-> 
( x  e.  B  /\  y  e.  B
) ) )
8 fveq2 5866 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
9 fveq2 5866 . . . . . . . 8  |-  ( w  =  W  ->  ( lt `  w )  =  ( lt `  W
) )
10 eqidd 2468 . . . . . . . 8  |-  ( w  =  W  ->  x  =  x )
118, 9, 10breq123d 4461 . . . . . . 7  |-  ( w  =  W  ->  (
( 0g `  w
) ( lt `  w ) x  <->  ( 0g `  W ) ( lt
`  W ) x ) )
12 fveq2 5866 . . . . . . . . . 10  |-  ( w  =  W  ->  (.g `  w )  =  (.g `  W ) )
1312oveqd 6302 . . . . . . . . 9  |-  ( w  =  W  ->  (
n (.g `  w ) x )  =  ( n (.g `  W ) x ) )
14 eqidd 2468 . . . . . . . . 9  |-  ( w  =  W  ->  y  =  y )
1513, 9, 14breq123d 4461 . . . . . . . 8  |-  ( w  =  W  ->  (
( n (.g `  w
) x ) ( lt `  w ) y  <->  ( n (.g `  W ) x ) ( lt `  W
) y ) )
1615ralbidv 2903 . . . . . . 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 710 . . . . . 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 710 . . . . 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 4510 . . . 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 27481 . . . 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 5876 . . . . . . 7  |-  ( Base `  W )  e.  _V
223, 21eqeltri 2551 . . . . . 6  |-  B  e. 
_V
2322, 22xpex 6589 . . . . 5  |-  ( B  X.  B )  e. 
_V
24 opabssxp 5074 . . . . 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 4592 . . . 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 5951 . . 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 16 . 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 3554 1  |-  ( W  e.  V  ->  (<<< `  W )  C_  ( B  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    C_ wss 3476   class class class wbr 4447   {copab 4504    X. cxp 4997   ` cfv 5588  (class class class)co 6285   NNcn 10537   Basecbs 14493   0gc0g 14698   ltcplt 15431  .gcmg 15734  <<<cinftm 27479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-inftm 27481
This theorem is referenced by:  isarchi  27485
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