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Theorem pw2f1o2 27109
Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 7215, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypothesis
Ref Expression
pw2f1o2.f  |-  F  =  ( x  e.  ( 2o  ^m  A ) 
|->  ( `' x " { 1o } ) )
Assertion
Ref Expression
pw2f1o2  |-  ( A  e.  V  ->  F : ( 2o  ^m  A ) -1-1-onto-> ~P A )
Distinct variable groups:    x, A    x, V
Allowed substitution hint:    F( x)

Proof of Theorem pw2f1o2
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pw2f1o2.f . . 3  |-  F  =  ( x  e.  ( 2o  ^m  A ) 
|->  ( `' x " { 1o } ) )
21pw2f1ocnv 27108 . 2  |-  ( A  e.  V  ->  ( F : ( 2o  ^m  A ) -1-1-onto-> ~P A  /\  `' F  =  ( y  e.  ~P A  |->  ( z  e.  A  |->  if ( z  e.  y ,  1o ,  (/) ) ) ) ) )
32simpld 446 1  |-  ( A  e.  V  ->  F : ( 2o  ^m  A ) -1-1-onto-> ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   (/)c0 3628   ifcif 3739   ~Pcpw 3799   {csn 3814    e. cmpt 4266   `'ccnv 4877   "cima 4881   -1-1-onto->wf1o 5453  (class class class)co 6081   1oc1o 6717   2oc2o 6718    ^m cmap 7018
This theorem is referenced by:  wepwsolem  27116  pwfi2f1o  27237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1o 6724  df-2o 6725  df-map 7020
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