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Theorem noreson 25615
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 25601 . . 3  |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
2 onin 4612 . . . . . . . 8  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( x  i^i  B
)  e.  On )
3 fresin 5612 . . . . . . . 8  |-  ( A : x --> { 1o ,  2o }  ->  ( A  |`  B ) : ( x  i^i  B
) --> { 1o ,  2o } )
4 feq2 5577 . . . . . . . . 9  |-  ( y  =  ( x  i^i 
B )  ->  (
( A  |`  B ) : y --> { 1o ,  2o }  <->  ( A  |`  B ) : ( x  i^i  B ) --> { 1o ,  2o } ) )
54rspcev 3052 . . . . . . . 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 464 . . . . . . 7  |-  ( ( ( x  e.  On  /\  B  e.  On )  /\  A : x --> { 1o ,  2o } )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
76an32s 780 . . . . . 6  |-  ( ( ( x  e.  On  /\  A : x --> { 1o ,  2o } )  /\  B  e.  On )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
87ex 424 . . . . 5  |-  ( ( x  e.  On  /\  A : x --> { 1o ,  2o } )  -> 
( B  e.  On  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } ) )
98rexlimiva 2825 . . . 4  |-  ( E. x  e.  On  A : x --> { 1o ,  2o }  ->  ( B  e.  On  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } ) )
109imp 419 . . 3  |-  ( ( E. x  e.  On  A : x --> { 1o ,  2o }  /\  B  e.  On )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
111, 10sylanb 459 . 2  |-  ( ( A  e.  No  /\  B  e.  On )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
12 elno 25601 . 2  |-  ( ( A  |`  B )  e.  No  <->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
1311, 12sylibr 204 1  |-  ( ( A  e.  No  /\  B  e.  On )  ->  ( A  |`  B )  e.  No )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   E.wrex 2706    i^i cin 3319   {cpr 3815   Oncon0 4581    |` cres 4880   -->wf 5450   1oc1o 6717   2oc2o 6718   Nocsur 25595
This theorem is referenced by:  sltres  25619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-no 25598
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