MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cusgraexilem1 Structured version   Unicode version

Theorem cusgraexilem1 24289
Description: Lemma 1 for cusgraexi 24291. (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 4637 . . . 4  |-  ( V  e.  W  ->  ~P V  e.  _V )
3 rabexg 4603 . . . 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 2559 . 2  |-  ( V  e.  W  ->  P  e.  _V )
6 resiexg 6731 . 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 1379    e. wcel 1767   {crab 2821   _Vcvv 3118   ~Pcpw 4016    _I cid 4796    |` cres 5007   ` cfv 5594   2c2 10597   #chash 12385
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-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-id 4801  df-xp 5011  df-rel 5012  df-res 5017
This theorem is referenced by:  cusgraexilem2  24290  cusgraexi  24291  cusgraexg  24292
  Copyright terms: Public domain W3C validator