MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isf34lem2 Structured version   Unicode version

Theorem isf34lem2 8785
Description: Lemma for isfin3-4 8794. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypothesis
Ref Expression
compss.a  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
Assertion
Ref Expression
isf34lem2  |-  ( A  e.  V  ->  F : ~P A --> ~P A
)
Distinct variable groups:    x, A    x, V
Allowed substitution hint:    F( x)

Proof of Theorem isf34lem2
StepHypRef Expression
1 difss 3570 . . . 4  |-  ( A 
\  x )  C_  A
2 elpw2g 4557 . . . 4  |-  ( A  e.  V  ->  (
( A  \  x
)  e.  ~P A  <->  ( A  \  x ) 
C_  A ) )
31, 2mpbiri 233 . . 3  |-  ( A  e.  V  ->  ( A  \  x )  e. 
~P A )
43adantr 463 . 2  |-  ( ( A  e.  V  /\  x  e.  ~P A
)  ->  ( A  \  x )  e.  ~P A )
5 compss.a . 2  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
64, 5fmptd 6033 1  |-  ( A  e.  V  ->  F : ~P A --> ~P A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    \ cdif 3411    C_ wss 3414   ~Pcpw 3955    |-> cmpt 4453   -->wf 5565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577
This theorem is referenced by:  isf34lem5  8790  isf34lem7  8791  isf34lem6  8792
  Copyright terms: Public domain W3C validator