Theorem pw2f1o2 37605

Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8067, 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.

Step	Hyp	Ref	Expression
1		pw2f1o2.f	. . 3 |- F = ( x e. ( 2o ^m A ) |-> ( `' x " { 1o } ) )
2	1	pw2f1ocnv 37604	. 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 , (/) ) ) ) ) )
3	2	simpld 475	1 |- ( A e. V -> F : ( 2o ^m A ) -1-1-onto-> ~P A )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {csn 4177   |-> cmpt 4729   `'ccnv 5113   "cima 5117   -1-1-onto->wf1o 5887  (class class class)co 6650   1oc1o 7553   2oc2o 7554   ^m cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-2o 7561  df-map 7859

This theorem is referenced by:  wepwsolem  37612  pwfi2f1o  37666