Theorem undefval 5557

Description: Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 5559 instead.

Hypothesis

Ref	Expression
undefval.1	|- S e. _V

Assertion

Ref	Expression
undefval	|- (Undef` S) = ~PU.S

Proof of Theorem undefval

Step	Hyp	Ref	Expression
1		undefval.1	. 2 |- S e. _V
2		unieq 3185	. . . 4 |- (s = S -> U.s = U.S)
3		pweq 3036	. . . 4 |- (U.s = U.S -> ~PU.s = ~PU.S)
4	2, 3	syl 12	. . 3 |- (s = S -> ~PU.s = ~PU.S)
5		df-undef 5556	. . 3 |- Undef = (s e. _V |-> ~PU.s)
6	1	uniex 3794	. . . 4 |- U.S e. _V
7	6	pwex 3487	. . 3 |- ~PU.S e. _V
8	4, 5, 7	fvmpt 5015	. 2 |- (S e. _V -> (Undef` S) = ~PU.S)
9	1, 8	ax-mp 7	1 |- (Undef` S) = ~PU.S

Syntax hints:   = wceq 1298   e. wcel 1300  _Vcvv 2292  ~Pcpw 3032  U.cuni 3177  ` cfv 3998  Undefcund 5554

This theorem is referenced by:  undefnel2 5558

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-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-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  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-fv 4014  df-mpt 5006  df-undef 5556

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