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Theorem undefval 7017
Description: Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 7019 instead. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
undefval  |-  ( S  e.  V  ->  ( Undef `  S )  =  ~P U. S )

Proof of Theorem undefval
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 elex 3127 . 2  |-  ( S  e.  V  ->  S  e.  _V )
2 uniexg 6592 . . 3  |-  ( S  e.  V  ->  U. S  e.  _V )
3 pwexg 4637 . . 3  |-  ( U. S  e.  _V  ->  ~P
U. S  e.  _V )
42, 3syl 16 . 2  |-  ( S  e.  V  ->  ~P U. S  e.  _V )
5 unieq 4259 . . . 4  |-  ( s  =  S  ->  U. s  =  U. S )
65pweqd 4021 . . 3  |-  ( s  =  S  ->  ~P U. s  =  ~P U. S )
7 df-undef 7014 . . 3  |-  Undef  =  ( s  e.  _V  |->  ~P
U. s )
86, 7fvmptg 5955 . 2  |-  ( ( S  e.  _V  /\  ~P U. S  e.  _V )  ->  ( Undef `  S
)  =  ~P U. S )
91, 4, 8syl2anc 661 1  |-  ( S  e.  V  ->  ( Undef `  S )  =  ~P U. S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3118   ~Pcpw 4016   U.cuni 4251   ` cfv 5594   Undefcund 7013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-undef 7014
This theorem is referenced by:  undefnel2  7018  undefne0  7020
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