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Theorem noreson 24385
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 24371 . . 3  |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
2 onin 4439 . . . . . . . 8  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( x  i^i  B
)  e.  On )
3 fresin 5426 . . . . . . . 8  |-  ( A : x --> { 1o ,  2o }  ->  ( A  |`  B ) : ( x  i^i  B
) --> { 1o ,  2o } )
4 feq2 5392 . . . . . . . . 9  |-  ( y  =  ( x  i^i 
B )  ->  (
( A  |`  B ) : y --> { 1o ,  2o }  <->  ( A  |`  B ) : ( x  i^i  B ) --> { 1o ,  2o } ) )
54rspcev 2897 . . . . . . . 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 2675 . . . 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 24371 . 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 1696   E.wrex 2557    i^i cin 3164   {cpr 3654   Oncon0 4408    |` cres 4707   -->wf 5267   1oc1o 6488   2oc2o 6489   Nocsur 24365
This theorem is referenced by:  sltres  24389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-no 24368
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