Theorem nofv 13998

Description: The function value of a surreal is either a sign or the empty set.

Assertion

Ref	Expression
nofv	|- (A e. No -> ((A` X) = (/) \/ (A` X) = 1o \/ (A` X) = 2o))

Proof of Theorem nofv

Step	Hyp	Ref	Expression
1		pm2.1 718	. . 3 |- (-. X e. dom A \/ X e. dom A)
2		ndmfv 4702	. . . . 5 |- (-. X e. dom A -> (A` X) = (/))
3	2	a1i 8	. . . 4 |- (A e. No -> (-. X e. dom A -> (A` X) = (/)))
4		nofun 13993	. . . . 5 |- (A e. No -> Fun A)
5		norn 13995	. . . . 5 |- (A e. No -> ran A C_ {1o, 2o})
6		ssel 2615	. . . . . . . . . 10 |- (ran A C_ {1o, 2o} -> ((A` X) e. ran A -> (A` X) e. {1o, 2o}))
7		fvelrn 4785	. . . . . . . . . 10 |- ((Fun A /\ X e. dom A) -> (A` X) e. ran A)
8	6, 7	syl5com 63	. . . . . . . . 9 |- ((Fun A /\ X e. dom A) -> (ran A C_ {1o, 2o} -> (A` X) e. {1o, 2o}))
9	8	ex 402	. . . . . . . 8 |- (Fun A -> (X e. dom A -> (ran A C_ {1o, 2o} -> (A` X) e. {1o, 2o})))
10	9	com23 36	. . . . . . 7 |- (Fun A -> (ran A C_ {1o, 2o} -> (X e. dom A -> (A` X) e. {1o, 2o})))
11	10	imp 377	. . . . . 6 |- ((Fun A /\ ran A C_ {1o, 2o}) -> (X e. dom A -> (A` X) e. {1o, 2o}))
12		1on 5182	. . . . . . . 8 |- 1o e. On
13	12	elisseti 2301	. . . . . . 7 |- 1o e. _V
14		2on 5183	. . . . . . . 8 |- 2o e. On
15	14	elisseti 2301	. . . . . . 7 |- 2o e. _V
16	13, 15	elpr2 3062	. . . . . 6 |- ((A` X) e. {1o, 2o} <-> ((A` X) = 1o \/ (A` X) = 2o))
17	11, 16	syl6ib 229	. . . . 5 |- ((Fun A /\ ran A C_ {1o, 2o}) -> (X e. dom A -> ((A` X) = 1o \/ (A` X) = 2o)))
18	4, 5, 17	syl11anc 524	. . . 4 |- (A e. No -> (X e. dom A -> ((A` X) = 1o \/ (A` X) = 2o)))
19	3, 18	orim12d 624	. . 3 |- (A e. No -> ((-. X e. dom A \/ X e. dom A) -> ((A` X) = (/) \/ ((A` X) = 1o \/ (A` X) = 2o))))
20	1, 19	mpi 55	. 2 |- (A e. No -> ((A` X) = (/) \/ ((A` X) = 1o \/ (A` X) = 2o)))
21		3orass 861	. 2 |- (((A` X) = (/) \/ (A` X) = 1o \/ (A` X) = 2o) <-> ((A` X) = (/) \/ ((A` X) = 1o \/ (A` X) = 2o)))
22	20, 21	sylibr 217	1 |- (A e. No -> ((A` X) = (/) \/ (A` X) = 1o \/ (A` X) = 2o))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240   \/ w3o 857   = wceq 1298   e. wcel 1300   C_ wss 2593  (/)c0 2875  {cpr 3045  Oncon0 3657  dom cdm 3986  ran crn 3987  Fun wfun 3992  ` cfv 3998  1oc1o 5172  2oc2o 5173   No csur 13981

This theorem is referenced by:  axfelem8 14038  axfelem9 14039

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-1o 5177  df-2o 5178  df-no 13984

Copyright terms: Public domain