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Theorem f1opw 5984
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 5380 . . . 4  |-  ( F : A -1-1-onto-> B  <->  ( F : A -onto-> B  /\  Fun  `' F ) )
32simprbi 273 . . 3  |-  ( F : A -1-1-onto-> B  ->  Fun  `' F
)
4 vex 2692 . . . 4  |-  a  e. 
_V
54funimaex 5215 . . 3  |-  ( Fun  `' F  ->  ( `' F " a )  e.  _V )
63, 5syl 14 . 2  |-  ( F : A -1-1-onto-> B  ->  ( `' F " a )  e. 
_V )
7 f1ofun 5376 . . 3  |-  ( F : A -1-1-onto-> B  ->  Fun  F )
8 vex 2692 . . . 4  |-  b  e. 
_V
98funimaex 5215 . . 3  |-  ( Fun 
F  ->  ( F " b )  e.  _V )
107, 9syl 14 . 2  |-  ( F : A -1-1-onto-> B  ->  ( F " b )  e.  _V )
111, 6, 10f1opw2 5983 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 1481   _Vcvv 2689   ~Pcpw 3514    |-> cmpt 3996   `'ccnv 4545   "cima 4549   Fun wfun 5124   -onto->wfo 5128   -1-1-onto->wf1o 5129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137
This theorem is referenced by: (None)
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