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Theorem nofv 29613
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 5896 . . . . 5  |-  ( -.  X  e.  dom  A  ->  ( A `  X
)  =  (/) )
32a1i 11 . . . 4  |-  ( A  e.  No  ->  ( -.  X  e.  dom  A  ->  ( A `  X )  =  (/) ) )
4 nofun 29605 . . . . 5  |-  ( A  e.  No  ->  Fun  A )
5 norn 29607 . . . . 5  |-  ( A  e.  No  ->  ran  A 
C_  { 1o ,  2o } )
6 fvelrn 6025 . . . . . . . 8  |-  ( ( Fun  A  /\  X  e.  dom  A )  -> 
( A `  X
)  e.  ran  A
)
7 ssel 3493 . . . . . . . 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 7155 . . . . . . . 8  |-  1o  e.  On
1110elexi 3119 . . . . . . 7  |-  1o  e.  _V
12 2on 7156 . . . . . . . 8  |-  2o  e.  On
1312elexi 3119 . . . . . . 7  |-  2o  e.  _V
1411, 13elpr2 4051 . . . . . 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 838 . . 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 976 . 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 972    = wceq 1395    e. wcel 1819    C_ wss 3471   (/)c0 3793   {cpr 4034   Oncon0 4887   dom cdm 5008   ran crn 5009   Fun wfun 5588   ` cfv 5594   1oc1o 7141   2oc2o 7142   Nocsur 29596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-1o 7148  df-2o 7149  df-no 29599
This theorem is referenced by:  nobndup  29656  nobnddown  29657
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