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Theorem f1opw 6301
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 21 . 2  |-  ( F : A -1-1-onto-> B  ->  F : A
-1-1-onto-> B )
2 dff1o3 5682 . . . 4  |-  ( F : A -1-1-onto-> B  <->  ( F : A -onto-> B  /\  Fun  `' F ) )
32simprbi 452 . . 3  |-  ( F : A -1-1-onto-> B  ->  Fun  `' F
)
4 vex 2961 . . . 4  |-  a  e. 
_V
54funimaex 5533 . . 3  |-  ( Fun  `' F  ->  ( `' F " a )  e.  _V )
63, 5syl 16 . 2  |-  ( F : A -1-1-onto-> B  ->  ( `' F " a )  e. 
_V )
7 f1ofun 5678 . . 3  |-  ( F : A -1-1-onto-> B  ->  Fun  F )
8 vex 2961 . . . 4  |-  b  e. 
_V
98funimaex 5533 . . 3  |-  ( Fun 
F  ->  ( F " b )  e.  _V )
107, 9syl 16 . 2  |-  ( F : A -1-1-onto-> B  ->  ( F " b )  e.  _V )
111, 6, 10f1opw2 6300 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 1726   _Vcvv 2958   ~Pcpw 3801    e. cmpt 4268   `'ccnv 4879   "cima 4883   Fun wfun 5450   -onto->wfo 5454   -1-1-onto->wf1o 5455
This theorem is referenced by:  ackbij2lem2  8122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463
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