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Theorem cusgraexilem1 23374
Description: Lemma 1 for cusgraexi 23376. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
Hypothesis
Ref Expression
cusgraexi.p  |-  P  =  { x  e.  ~P V  |  ( # `  x
)  =  2 }
Assertion
Ref Expression
cusgraexilem1  |-  ( V  e.  W  ->  (  _I  |`  P )  e. 
_V )
Distinct variable group:    x, V
Allowed substitution hints:    P( x)    W( x)

Proof of Theorem cusgraexilem1
StepHypRef Expression
1 cusgraexi.p . . 3  |-  P  =  { x  e.  ~P V  |  ( # `  x
)  =  2 }
2 pwexg 4476 . . . 4  |-  ( V  e.  W  ->  ~P V  e.  _V )
3 rabexg 4442 . . . 4  |-  ( ~P V  e.  _V  ->  { x  e.  ~P V  |  ( # `  x
)  =  2 }  e.  _V )
42, 3syl 16 . . 3  |-  ( V  e.  W  ->  { x  e.  ~P V  |  (
# `  x )  =  2 }  e.  _V )
51, 4syl5eqel 2527 . 2  |-  ( V  e.  W  ->  P  e.  _V )
6 resiexg 6514 . 2  |-  ( P  e.  _V  ->  (  _I  |`  P )  e. 
_V )
75, 6syl 16 1  |-  ( V  e.  W  ->  (  _I  |`  P )  e. 
_V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   {crab 2719   _Vcvv 2972   ~Pcpw 3860    _I cid 4631    |` cres 4842   ` cfv 5418   2c2 10371   #chash 12103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-res 4852
This theorem is referenced by:  cusgraexilem2  23375  cusgraexi  23376  cusgraexg  23377
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