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Theorem undefval 6301
Description: Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 6303 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 2796 . 2  |-  ( S  e.  V  ->  S  e.  _V )
2 uniexg 4517 . . 3  |-  ( S  e.  V  ->  U. S  e.  _V )
3 pwexg 4194 . . 3  |-  ( U. S  e.  _V  ->  ~P
U. S  e.  _V )
42, 3syl 15 . 2  |-  ( S  e.  V  ->  ~P U. S  e.  _V )
5 unieq 3836 . . . 4  |-  ( s  =  S  ->  U. s  =  U. S )
65pweqd 3630 . . 3  |-  ( s  =  S  ->  ~P U. s  =  ~P U. S )
7 df-undef 6298 . . 3  |-  Undef  =  ( s  e.  _V  |->  ~P
U. s )
86, 7fvmptg 5600 . 2  |-  ( ( S  e.  _V  /\  ~P U. S  e.  _V )  ->  ( Undef `  S
)  =  ~P U. S )
91, 4, 8syl2anc 642 1  |-  ( S  e.  V  ->  ( Undef `  S )  =  ~P U. S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   ~Pcpw 3625   U.cuni 3827   ` cfv 5255   Undefcund 6296
This theorem is referenced by:  undefnel2  6302  riotassuni  6343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-undef 6298
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