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Theorem undefval 6548
Description: Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 6550 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 2966 . 2  |-  ( S  e.  V  ->  S  e.  _V )
2 uniexg 4708 . . 3  |-  ( S  e.  V  ->  U. S  e.  _V )
3 pwexg 4385 . . 3  |-  ( U. S  e.  _V  ->  ~P
U. S  e.  _V )
42, 3syl 16 . 2  |-  ( S  e.  V  ->  ~P U. S  e.  _V )
5 unieq 4026 . . . 4  |-  ( s  =  S  ->  U. s  =  U. S )
65pweqd 3806 . . 3  |-  ( s  =  S  ->  ~P U. s  =  ~P U. S )
7 df-undef 6545 . . 3  |-  Undef  =  ( s  e.  _V  |->  ~P
U. s )
86, 7fvmptg 5806 . 2  |-  ( ( S  e.  _V  /\  ~P U. S  e.  _V )  ->  ( Undef `  S
)  =  ~P U. S )
91, 4, 8syl2anc 644 1  |-  ( S  e.  V  ->  ( Undef `  S )  =  ~P U. S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958   ~Pcpw 3801   U.cuni 4017   ` cfv 5456   Undefcund 6543
This theorem is referenced by:  undefnel2  6549  riotassuni  6590
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-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-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-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-undef 6545
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