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Theorem noreson 30334
Description: The restriction of a surreal to an ordinal is still a surreal. (Contributed by Scott Fenton, 4-Sep-2011.)
Assertion
Ref Expression
noreson  |-  ( ( A  e.  No  /\  B  e.  On )  ->  ( A  |`  B )  e.  No )

Proof of Theorem noreson
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elno 30320 . . 3  |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
2 onin 5473 . . . . . . . 8  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( x  i^i  B
)  e.  On )
3 fresin 5769 . . . . . . . 8  |-  ( A : x --> { 1o ,  2o }  ->  ( A  |`  B ) : ( x  i^i  B
) --> { 1o ,  2o } )
4 feq2 5729 . . . . . . . . 9  |-  ( y  =  ( x  i^i 
B )  ->  (
( A  |`  B ) : y --> { 1o ,  2o }  <->  ( A  |`  B ) : ( x  i^i  B ) --> { 1o ,  2o } ) )
54rspcev 3188 . . . . . . . 8  |-  ( ( ( x  i^i  B
)  e.  On  /\  ( A  |`  B ) : ( x  i^i 
B ) --> { 1o ,  2o } )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
62, 3, 5syl2an 479 . . . . . . 7  |-  ( ( ( x  e.  On  /\  B  e.  On )  /\  A : x --> { 1o ,  2o } )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
76an32s 811 . . . . . 6  |-  ( ( ( x  e.  On  /\  A : x --> { 1o ,  2o } )  /\  B  e.  On )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
87ex 435 . . . . 5  |-  ( ( x  e.  On  /\  A : x --> { 1o ,  2o } )  -> 
( B  e.  On  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } ) )
98rexlimiva 2920 . . . 4  |-  ( E. x  e.  On  A : x --> { 1o ,  2o }  ->  ( B  e.  On  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } ) )
109imp 430 . . 3  |-  ( ( E. x  e.  On  A : x --> { 1o ,  2o }  /\  B  e.  On )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
111, 10sylanb 474 . 2  |-  ( ( A  e.  No  /\  B  e.  On )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
12 elno 30320 . 2  |-  ( ( A  |`  B )  e.  No  <->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
1311, 12sylibr 215 1  |-  ( ( A  e.  No  /\  B  e.  On )  ->  ( A  |`  B )  e.  No )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1870   E.wrex 2783    i^i cin 3441   {cpr 4004    |` cres 4856   Oncon0 5442   -->wf 5597   1oc1o 7183   2oc2o 7184   Nocsur 30314
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 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661
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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-no 30317
This theorem is referenced by:  sltres  30338
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