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Theorem f1opw 6088
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 5494 . . . 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 2804 . . . 4  |-  a  e. 
_V
54funimaex 5346 . . 3  |-  ( Fun  `' F  ->  ( `' F " a )  e.  _V )
63, 5syl 15 . 2  |-  ( F : A -1-1-onto-> B  ->  ( `' F " a )  e. 
_V )
7 f1ofun 5490 . . 3  |-  ( F : A -1-1-onto-> B  ->  Fun  F )
8 vex 2804 . . . 4  |-  b  e. 
_V
98funimaex 5346 . . 3  |-  ( Fun 
F  ->  ( F " b )  e.  _V )
107, 9syl 15 . 2  |-  ( F : A -1-1-onto-> B  ->  ( F " b )  e.  _V )
111, 6, 10f1opw2 6087 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 1696   _Vcvv 2801   ~Pcpw 3638    e. cmpt 4093   `'ccnv 4704   "cima 4708   Fun wfun 5265   -onto->wfo 5269   -1-1-onto->wf1o 5270
This theorem is referenced by:  ackbij2lem2  7882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278
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