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Theorem nofv 27796
Description: The function value of a surreal is either a sign or the empty set. (Contributed by Scott Fenton, 22-Jun-2011.)
Assertion
Ref Expression
nofv  |-  ( A  e.  No  ->  (
( A `  X
)  =  (/)  \/  ( A `  X )  =  1o  \/  ( A `  X )  =  2o ) )

Proof of Theorem nofv
StepHypRef Expression
1 pm2.1 417 . . 3  |-  ( -.  X  e.  dom  A  \/  X  e.  dom  A )
2 ndmfv 5712 . . . . 5  |-  ( -.  X  e.  dom  A  ->  ( A `  X
)  =  (/) )
32a1i 11 . . . 4  |-  ( A  e.  No  ->  ( -.  X  e.  dom  A  ->  ( A `  X )  =  (/) ) )
4 nofun 27788 . . . . 5  |-  ( A  e.  No  ->  Fun  A )
5 norn 27790 . . . . 5  |-  ( A  e.  No  ->  ran  A 
C_  { 1o ,  2o } )
6 fvelrn 5837 . . . . . . . 8  |-  ( ( Fun  A  /\  X  e.  dom  A )  -> 
( A `  X
)  e.  ran  A
)
7 ssel 3348 . . . . . . . 8  |-  ( ran 
A  C_  { 1o ,  2o }  ->  (
( A `  X
)  e.  ran  A  ->  ( A `  X
)  e.  { 1o ,  2o } ) )
86, 7syl5com 30 . . . . . . 7  |-  ( ( Fun  A  /\  X  e.  dom  A )  -> 
( ran  A  C_  { 1o ,  2o }  ->  ( A `  X )  e.  { 1o ,  2o } ) )
98impancom 440 . . . . . 6  |-  ( ( Fun  A  /\  ran  A 
C_  { 1o ,  2o } )  ->  ( X  e.  dom  A  -> 
( A `  X
)  e.  { 1o ,  2o } ) )
10 1on 6925 . . . . . . . 8  |-  1o  e.  On
1110elexi 2980 . . . . . . 7  |-  1o  e.  _V
12 2on 6926 . . . . . . . 8  |-  2o  e.  On
1312elexi 2980 . . . . . . 7  |-  2o  e.  _V
1411, 13elpr2 3894 . . . . . 6  |-  ( ( A `  X )  e.  { 1o ,  2o }  <->  ( ( A `
 X )  =  1o  \/  ( A `
 X )  =  2o ) )
159, 14syl6ib 226 . . . . 5  |-  ( ( Fun  A  /\  ran  A 
C_  { 1o ,  2o } )  ->  ( X  e.  dom  A  -> 
( ( A `  X )  =  1o  \/  ( A `  X )  =  2o ) ) )
164, 5, 15syl2anc 661 . . . 4  |-  ( A  e.  No  ->  ( X  e.  dom  A  -> 
( ( A `  X )  =  1o  \/  ( A `  X )  =  2o ) ) )
173, 16orim12d 834 . . 3  |-  ( A  e.  No  ->  (
( -.  X  e. 
dom  A  \/  X  e.  dom  A )  -> 
( ( A `  X )  =  (/)  \/  ( ( A `  X )  =  1o  \/  ( A `  X )  =  2o ) ) ) )
181, 17mpi 17 . 2  |-  ( A  e.  No  ->  (
( A `  X
)  =  (/)  \/  (
( A `  X
)  =  1o  \/  ( A `  X )  =  2o ) ) )
19 3orass 968 . 2  |-  ( ( ( A `  X
)  =  (/)  \/  ( A `  X )  =  1o  \/  ( A `  X )  =  2o )  <->  ( ( A `  X )  =  (/)  \/  ( ( A `  X )  =  1o  \/  ( A `  X )  =  2o ) ) )
2018, 19sylibr 212 1  |-  ( A  e.  No  ->  (
( A `  X
)  =  (/)  \/  ( A `  X )  =  1o  \/  ( A `  X )  =  2o ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 964    = wceq 1369    e. wcel 1756    C_ wss 3326   (/)c0 3635   {cpr 3877   Oncon0 4717   dom cdm 4838   ran crn 4839   Fun wfun 5410   ` cfv 5416   1oc1o 6911   2oc2o 6912   Nocsur 27779
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-1o 6918  df-2o 6919  df-no 27782
This theorem is referenced by:  nobndup  27839  nobnddown  27840
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