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Theorem nofv 28980
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 5881 . . . . 5  |-  ( -.  X  e.  dom  A  ->  ( A `  X
)  =  (/) )
32a1i 11 . . . 4  |-  ( A  e.  No  ->  ( -.  X  e.  dom  A  ->  ( A `  X )  =  (/) ) )
4 nofun 28972 . . . . 5  |-  ( A  e.  No  ->  Fun  A )
5 norn 28974 . . . . 5  |-  ( A  e.  No  ->  ran  A 
C_  { 1o ,  2o } )
6 fvelrn 6008 . . . . . . . 8  |-  ( ( Fun  A  /\  X  e.  dom  A )  -> 
( A `  X
)  e.  ran  A
)
7 ssel 3491 . . . . . . . 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 7127 . . . . . . . 8  |-  1o  e.  On
1110elexi 3116 . . . . . . 7  |-  1o  e.  _V
12 2on 7128 . . . . . . . 8  |-  2o  e.  On
1312elexi 3116 . . . . . . 7  |-  2o  e.  _V
1411, 13elpr2 4039 . . . . . 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 835 . . 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 971 . 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 967    = wceq 1374    e. wcel 1762    C_ wss 3469   (/)c0 3778   {cpr 4022   Oncon0 4871   dom cdm 4992   ran crn 4993   Fun wfun 5573   ` cfv 5579   1oc1o 7113   2oc2o 7114   Nocsur 28963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-1o 7120  df-2o 7121  df-no 28966
This theorem is referenced by:  nobndup  29023  nobnddown  29024
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