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Theorem pw2f1o2 27131
Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 6969, 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 27130 . 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 445 1  |-  ( A  e.  V  ->  F : ( 2o  ^m  A ) -1-1-onto-> ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   (/)c0 3455   ifcif 3565   ~Pcpw 3625   {csn 3640    e. cmpt 4077   `'ccnv 4688   "cima 4692   -1-1-onto->wf1o 5254  (class class class)co 5858   1oc1o 6472   2oc2o 6473    ^m cmap 6772
This theorem is referenced by:  wepwsolem  27138  pwfi2f1o  27260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1o 6479  df-2o 6480  df-map 6774
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