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Theorem f1opw 6072
Description: A one-to-one mapping induces a one-to-one mapping on power sets. (Contributed by Stefan O'Rear, 18-Nov-2014.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
f1opw  |-  ( F : A -1-1-onto-> B  ->  ( b  e.  ~P A  |->  ( F
" b ) ) : ~P A -1-1-onto-> ~P B
)
Distinct variable groups:    A, b    B, b    F, b

Proof of Theorem f1opw
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 id 19 . 2  |-  ( F : A -1-1-onto-> B  ->  F : A
-1-1-onto-> B )
2 dff1o3 5478 . . . 4  |-  ( F : A -1-1-onto-> B  <->  ( F : A -onto-> B  /\  Fun  `' F ) )
32simprbi 450 . . 3  |-  ( F : A -1-1-onto-> B  ->  Fun  `' F
)
4 vex 2791 . . . 4  |-  a  e. 
_V
54funimaex 5330 . . 3  |-  ( Fun  `' F  ->  ( `' F " a )  e.  _V )
63, 5syl 15 . 2  |-  ( F : A -1-1-onto-> B  ->  ( `' F " a )  e. 
_V )
7 f1ofun 5474 . . 3  |-  ( F : A -1-1-onto-> B  ->  Fun  F )
8 vex 2791 . . . 4  |-  b  e. 
_V
98funimaex 5330 . . 3  |-  ( Fun 
F  ->  ( F " b )  e.  _V )
107, 9syl 15 . 2  |-  ( F : A -1-1-onto-> B  ->  ( F " b )  e.  _V )
111, 6, 10f1opw2 6071 1  |-  ( F : A -1-1-onto-> B  ->  ( b  e.  ~P A  |->  ( F
" b ) ) : ~P A -1-1-onto-> ~P B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   _Vcvv 2788   ~Pcpw 3625    e. cmpt 4077   `'ccnv 4688   "cima 4692   Fun wfun 5249   -onto->wfo 5253   -1-1-onto->wf1o 5254
This theorem is referenced by:  ackbij2lem2  7866
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-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-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262
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