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Theorem inftmrel 28509
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 3056 . . 3  |-  ( W  e.  V  ->  W  e.  _V )
2 fveq2 5870 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
3 inftm.b . . . . . . . . 9  |-  B  =  ( Base `  W
)
42, 3syl6eqr 2505 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  B )
54eleq2d 2516 . . . . . . 7  |-  ( w  =  W  ->  (
x  e.  ( Base `  w )  <->  x  e.  B ) )
64eleq2d 2516 . . . . . . 7  |-  ( w  =  W  ->  (
y  e.  ( Base `  w )  <->  y  e.  B ) )
75, 6anbi12d 718 . . . . . 6  |-  ( w  =  W  ->  (
( x  e.  (
Base `  w )  /\  y  e.  ( Base `  w ) )  <-> 
( x  e.  B  /\  y  e.  B
) ) )
8 fveq2 5870 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
9 fveq2 5870 . . . . . . . 8  |-  ( w  =  W  ->  ( lt `  w )  =  ( lt `  W
) )
10 eqidd 2454 . . . . . . . 8  |-  ( w  =  W  ->  x  =  x )
118, 9, 10breq123d 4419 . . . . . . 7  |-  ( w  =  W  ->  (
( 0g `  w
) ( lt `  w ) x  <->  ( 0g `  W ) ( lt
`  W ) x ) )
12 fveq2 5870 . . . . . . . . . 10  |-  ( w  =  W  ->  (.g `  w )  =  (.g `  W ) )
1312oveqd 6312 . . . . . . . . 9  |-  ( w  =  W  ->  (
n (.g `  w ) x )  =  ( n (.g `  W ) x ) )
14 eqidd 2454 . . . . . . . . 9  |-  ( w  =  W  ->  y  =  y )
1513, 9, 14breq123d 4419 . . . . . . . 8  |-  ( w  =  W  ->  (
( n (.g `  w
) x ) ( lt `  w ) y  <->  ( n (.g `  W ) x ) ( lt `  W
) y ) )
1615ralbidv 2829 . . . . . . 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 718 . . . . . 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 718 . . . . 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 4469 . . . 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 28507 . . . 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 5880 . . . . . . 7  |-  ( Base `  W )  e.  _V
223, 21eqeltri 2527 . . . . . 6  |-  B  e. 
_V
2322, 22xpex 6600 . . . . 5  |-  ( B  X.  B )  e. 
_V
24 opabssxp 4912 . . . . 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 4551 . . . 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 5953 . . 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 3484 1  |-  ( W  e.  V  ->  (<<< `  W )  C_  ( B  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1446    e. wcel 1889   A.wral 2739   _Vcvv 3047    C_ wss 3406   class class class wbr 4405   {copab 4463    X. cxp 4835   ` cfv 5585  (class class class)co 6295   NNcn 10616   Basecbs 15133   0gc0g 15350   ltcplt 16198  .gcmg 16684  <<<cinftm 28505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5549  df-fun 5587  df-fv 5593  df-ov 6298  df-inftm 28507
This theorem is referenced by:  isarchi  28511
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