Theorem f1opw 6889

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.

Step	Hyp	Ref	Expression
1		id 22	. 2 |- ( F : A -1-1-onto-> B -> F : A -1-1-onto-> B )
2		dff1o3 6143	. . . 4 |- ( F : A -1-1-onto-> B <-> ( F : A -onto-> B /\ Fun `' F ) )
3	2	simprbi 480	. . 3 |- ( F : A -1-1-onto-> B -> Fun `' F )
4		vex 3203	. . . 4 |- a e. _V
5	4	funimaex 5976	. . 3 |- ( Fun `' F -> ( `' F " a ) e. _V )
6	3, 5	syl 17	. 2 |- ( F : A -1-1-onto-> B -> ( `' F " a ) e. _V )
7		f1ofun 6139	. . 3 |- ( F : A -1-1-onto-> B -> Fun F )
8		vex 3203	. . . 4 |- b e. _V
9	8	funimaex 5976	. . 3 |- ( Fun F -> ( F " b ) e. _V )
10	7, 9	syl 17	. 2 |- ( F : A -1-1-onto-> B -> ( F " b ) e. _V )
11	1, 6, 10	f1opw2 6888	1 |- ( F : A -1-1-onto-> B -> ( b e. ~P A |-> ( F " b ) ) : ~P A -1-1-onto-> ~P B )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   ~Pcpw 4158   |-> cmpt 4729   `'ccnv 5113   "cima 5117   Fun wfun 5882   -onto->wfo 5886   -1-1-onto->wf1o 5887

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-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-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  ackbij2lem2  9062