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Theorem nofv 25614
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 pm2.1 408 . . 3  |-  ( -.  X  e.  dom  A  \/  X  e.  dom  A )
2 ndmfv 5757 . . . . 5  |-  ( -.  X  e.  dom  A  ->  ( A `  X
)  =  (/) )
32a1i 11 . . . 4  |-  ( A  e.  No  ->  ( -.  X  e.  dom  A  ->  ( A `  X )  =  (/) ) )
4 nofun 25606 . . . . 5  |-  ( A  e.  No  ->  Fun  A )
5 norn 25608 . . . . 5  |-  ( A  e.  No  ->  ran  A 
C_  { 1o ,  2o } )
6 fvelrn 5868 . . . . . . . 8  |-  ( ( Fun  A  /\  X  e.  dom  A )  -> 
( A `  X
)  e.  ran  A
)
7 ssel 3344 . . . . . . . 8  |-  ( ran 
A  C_  { 1o ,  2o }  ->  (
( A `  X
)  e.  ran  A  ->  ( A `  X
)  e.  { 1o ,  2o } ) )
86, 7syl5com 29 . . . . . . 7  |-  ( ( Fun  A  /\  X  e.  dom  A )  -> 
( ran  A  C_  { 1o ,  2o }  ->  ( A `  X )  e.  { 1o ,  2o } ) )
98impancom 429 . . . . . 6  |-  ( ( Fun  A  /\  ran  A 
C_  { 1o ,  2o } )  ->  ( X  e.  dom  A  -> 
( A `  X
)  e.  { 1o ,  2o } ) )
10 1on 6733 . . . . . . . 8  |-  1o  e.  On
1110elexi 2967 . . . . . . 7  |-  1o  e.  _V
12 2on 6734 . . . . . . . 8  |-  2o  e.  On
1312elexi 2967 . . . . . . 7  |-  2o  e.  _V
1411, 13elpr2 3835 . . . . . 6  |-  ( ( A `  X )  e.  { 1o ,  2o }  <->  ( ( A `
 X )  =  1o  \/  ( A `
 X )  =  2o ) )
159, 14syl6ib 219 . . . . 5  |-  ( ( Fun  A  /\  ran  A 
C_  { 1o ,  2o } )  ->  ( X  e.  dom  A  -> 
( ( A `  X )  =  1o  \/  ( A `  X )  =  2o ) ) )
164, 5, 15syl2anc 644 . . . 4  |-  ( A  e.  No  ->  ( X  e.  dom  A  -> 
( ( A `  X )  =  1o  \/  ( A `  X )  =  2o ) ) )
173, 16orim12d 813 . . 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 940 . 2  |-  ( ( ( A `  X
)  =  (/)  \/  ( A `  X )  =  1o  \/  ( A `  X )  =  2o )  <->  ( ( A `  X )  =  (/)  \/  ( ( A `  X )  =  1o  \/  ( A `  X )  =  2o ) ) )
2018, 19sylibr 205 1  |-  ( A  e.  No  ->  (
( A `  X
)  =  (/)  \/  ( A `  X )  =  1o  \/  ( A `  X )  =  2o ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 359    /\ wa 360    \/ w3o 936    = wceq 1653    e. wcel 1726    C_ wss 3322   (/)c0 3630   {cpr 3817   Oncon0 4583   dom cdm 4880   ran crn 4881   Fun wfun 5450   ` cfv 5456   1oc1o 6719   2oc2o 6720   Nocsur 25597
This theorem is referenced by:  nobndup  25657  nobnddown  25658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-suc 4589  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1o 6726  df-2o 6727  df-no 25600
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