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Theorem noreson 24314
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 24300 . . 3  |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
2 onin 4423 . . . . . . . 8  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( x  i^i  B
)  e.  On )
3 fresin 5410 . . . . . . . 8  |-  ( A : x --> { 1o ,  2o }  ->  ( A  |`  B ) : ( x  i^i  B
) --> { 1o ,  2o } )
4 feq2 5376 . . . . . . . . 9  |-  ( y  =  ( x  i^i 
B )  ->  (
( A  |`  B ) : y --> { 1o ,  2o }  <->  ( A  |`  B ) : ( x  i^i  B ) --> { 1o ,  2o } ) )
54rspcev 2884 . . . . . . . 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 463 . . . . . . 7  |-  ( ( ( x  e.  On  /\  B  e.  On )  /\  A : x --> { 1o ,  2o } )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
76an32s 779 . . . . . 6  |-  ( ( ( x  e.  On  /\  A : x --> { 1o ,  2o } )  /\  B  e.  On )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
87ex 423 . . . . 5  |-  ( ( x  e.  On  /\  A : x --> { 1o ,  2o } )  -> 
( B  e.  On  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } ) )
98rexlimiva 2662 . . . 4  |-  ( E. x  e.  On  A : x --> { 1o ,  2o }  ->  ( B  e.  On  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } ) )
109imp 418 . . 3  |-  ( ( E. x  e.  On  A : x --> { 1o ,  2o }  /\  B  e.  On )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
111, 10sylanb 458 . 2  |-  ( ( A  e.  No  /\  B  e.  On )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
12 elno 24300 . 2  |-  ( ( A  |`  B )  e.  No  <->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
1311, 12sylibr 203 1  |-  ( ( A  e.  No  /\  B  e.  On )  ->  ( A  |`  B )  e.  No )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   E.wrex 2544    i^i cin 3151   {cpr 3641   Oncon0 4392    |` cres 4691   -->wf 5251   1oc1o 6472   2oc2o 6473   Nocsur 24294
This theorem is referenced by:  sltres  24318
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-no 24297
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