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Theorem inftmrel 26209
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 2993 . . 3  |-  ( W  e.  V  ->  W  e.  _V )
2 fveq2 5703 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
3 inftm.b . . . . . . . . 9  |-  B  =  ( Base `  W
)
42, 3syl6eqr 2493 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  B )
54eleq2d 2510 . . . . . . 7  |-  ( w  =  W  ->  (
x  e.  ( Base `  w )  <->  x  e.  B ) )
64eleq2d 2510 . . . . . . 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 5703 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
9 fveq2 5703 . . . . . . . 8  |-  ( w  =  W  ->  ( lt `  w )  =  ( lt `  W
) )
10 eqidd 2444 . . . . . . . 8  |-  ( w  =  W  ->  x  =  x )
118, 9, 10breq123d 4318 . . . . . . 7  |-  ( w  =  W  ->  (
( 0g `  w
) ( lt `  w ) x  <->  ( 0g `  W ) ( lt
`  W ) x ) )
12 fveq2 5703 . . . . . . . . . 10  |-  ( w  =  W  ->  (.g `  w )  =  (.g `  W ) )
1312oveqd 6120 . . . . . . . . 9  |-  ( w  =  W  ->  (
n (.g `  w ) x )  =  ( n (.g `  W ) x ) )
14 eqidd 2444 . . . . . . . . 9  |-  ( w  =  W  ->  y  =  y )
1513, 9, 14breq123d 4318 . . . . . . . 8  |-  ( w  =  W  ->  (
( n (.g `  w
) x ) ( lt `  w ) y  <->  ( n (.g `  W ) x ) ( lt `  W
) y ) )
1615ralbidv 2747 . . . . . . 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 4367 . . . 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 26207 . . . 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 5713 . . . . . . 7  |-  ( Base `  W )  e.  _V
223, 21eqeltri 2513 . . . . . 6  |-  B  e. 
_V
2322, 22xpex 6520 . . . . 5  |-  ( B  X.  B )  e. 
_V
24 opabssxp 4923 . . . . 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 4449 . . . 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 5786 . . 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 3418 1  |-  ( W  e.  V  ->  (<<< `  W )  C_  ( B  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2727   _Vcvv 2984    C_ wss 3340   class class class wbr 4304   {copab 4361    X. cxp 4850   ` cfv 5430  (class class class)co 6103   NNcn 10334   Basecbs 14186   0gc0g 14390   ltcplt 15123  .gcmg 15426  <<<cinftm 26205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5393  df-fun 5432  df-fv 5438  df-ov 6106  df-inftm 26207
This theorem is referenced by:  isarchi  26211
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