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Theorem noreson 27806
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 27792 . . 3  |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
2 onin 4755 . . . . . . . 8  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( x  i^i  B
)  e.  On )
3 fresin 5585 . . . . . . . 8  |-  ( A : x --> { 1o ,  2o }  ->  ( A  |`  B ) : ( x  i^i  B
) --> { 1o ,  2o } )
4 feq2 5548 . . . . . . . . 9  |-  ( y  =  ( x  i^i 
B )  ->  (
( A  |`  B ) : y --> { 1o ,  2o }  <->  ( A  |`  B ) : ( x  i^i  B ) --> { 1o ,  2o } ) )
54rspcev 3078 . . . . . . . 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 477 . . . . . . 7  |-  ( ( ( x  e.  On  /\  B  e.  On )  /\  A : x --> { 1o ,  2o } )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
76an32s 802 . . . . . 6  |-  ( ( ( x  e.  On  /\  A : x --> { 1o ,  2o } )  /\  B  e.  On )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
87ex 434 . . . . 5  |-  ( ( x  e.  On  /\  A : x --> { 1o ,  2o } )  -> 
( B  e.  On  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } ) )
98rexlimiva 2841 . . . 4  |-  ( E. x  e.  On  A : x --> { 1o ,  2o }  ->  ( B  e.  On  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } ) )
109imp 429 . . 3  |-  ( ( E. x  e.  On  A : x --> { 1o ,  2o }  /\  B  e.  On )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
111, 10sylanb 472 . 2  |-  ( ( A  e.  No  /\  B  e.  On )  ->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
12 elno 27792 . 2  |-  ( ( A  |`  B )  e.  No  <->  E. y  e.  On  ( A  |`  B ) : y --> { 1o ,  2o } )
1311, 12sylibr 212 1  |-  ( ( A  e.  No  /\  B  e.  On )  ->  ( A  |`  B )  e.  No )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   E.wrex 2721    i^i cin 3332   {cpr 3884   Oncon0 4724    |` cres 4847   -->wf 5419   1oc1o 6918   2oc2o 6919   Nocsur 27786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-no 27789
This theorem is referenced by:  sltres  27810
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