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Theorem f1opw2 6510
Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6511 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
f1opw2.1  |-  ( ph  ->  F : A -1-1-onto-> B )
f1opw2.2  |-  ( ph  ->  ( `' F "
a )  e.  _V )
f1opw2.3  |-  ( ph  ->  ( F " b
)  e.  _V )
Assertion
Ref Expression
f1opw2  |-  ( ph  ->  ( b  e.  ~P A  |->  ( F "
b ) ) : ~P A -1-1-onto-> ~P B )
Distinct variable groups:    a, b, A    B, a, b    F, a, b    ph, a, b

Proof of Theorem f1opw2
StepHypRef Expression
1 eqid 2467 . 2  |-  ( b  e.  ~P A  |->  ( F " b ) )  =  ( b  e.  ~P A  |->  ( F " b ) )
2 imassrn 5346 . . . . 5  |-  ( F
" b )  C_  ran  F
3 f1opw2.1 . . . . . . 7  |-  ( ph  ->  F : A -1-1-onto-> B )
4 f1ofo 5821 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
53, 4syl 16 . . . . . 6  |-  ( ph  ->  F : A -onto-> B
)
6 forn 5796 . . . . . 6  |-  ( F : A -onto-> B  ->  ran  F  =  B )
75, 6syl 16 . . . . 5  |-  ( ph  ->  ran  F  =  B )
82, 7syl5sseq 3552 . . . 4  |-  ( ph  ->  ( F " b
)  C_  B )
9 f1opw2.3 . . . . 5  |-  ( ph  ->  ( F " b
)  e.  _V )
10 elpwg 4018 . . . . 5  |-  ( ( F " b )  e.  _V  ->  (
( F " b
)  e.  ~P B  <->  ( F " b ) 
C_  B ) )
119, 10syl 16 . . . 4  |-  ( ph  ->  ( ( F "
b )  e.  ~P B 
<->  ( F " b
)  C_  B )
)
128, 11mpbird 232 . . 3  |-  ( ph  ->  ( F " b
)  e.  ~P B
)
1312adantr 465 . 2  |-  ( (
ph  /\  b  e.  ~P A )  ->  ( F " b )  e. 
~P B )
14 imassrn 5346 . . . . 5  |-  ( `' F " a ) 
C_  ran  `' F
15 dfdm4 5193 . . . . . 6  |-  dom  F  =  ran  `' F
16 f1odm 5818 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  dom  F  =  A )
173, 16syl 16 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
1815, 17syl5eqr 2522 . . . . 5  |-  ( ph  ->  ran  `' F  =  A )
1914, 18syl5sseq 3552 . . . 4  |-  ( ph  ->  ( `' F "
a )  C_  A
)
20 f1opw2.2 . . . . 5  |-  ( ph  ->  ( `' F "
a )  e.  _V )
21 elpwg 4018 . . . . 5  |-  ( ( `' F " a )  e.  _V  ->  (
( `' F "
a )  e.  ~P A 
<->  ( `' F "
a )  C_  A
) )
2220, 21syl 16 . . . 4  |-  ( ph  ->  ( ( `' F " a )  e.  ~P A 
<->  ( `' F "
a )  C_  A
) )
2319, 22mpbird 232 . . 3  |-  ( ph  ->  ( `' F "
a )  e.  ~P A )
2423adantr 465 . 2  |-  ( (
ph  /\  a  e.  ~P B )  ->  ( `' F " a )  e.  ~P A )
25 elpwi 4019 . . . . . . 7  |-  ( a  e.  ~P B  -> 
a  C_  B )
2625adantl 466 . . . . . 6  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  a  C_  B )
27 foimacnv 5831 . . . . . 6  |-  ( ( F : A -onto-> B  /\  a  C_  B )  ->  ( F "
( `' F "
a ) )  =  a )
285, 26, 27syl2an 477 . . . . 5  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( F "
( `' F "
a ) )  =  a )
2928eqcomd 2475 . . . 4  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  a  =  ( F " ( `' F " a ) ) )
30 imaeq2 5331 . . . . 5  |-  ( b  =  ( `' F " a )  ->  ( F " b )  =  ( F " ( `' F " a ) ) )
3130eqeq2d 2481 . . . 4  |-  ( b  =  ( `' F " a )  ->  (
a  =  ( F
" b )  <->  a  =  ( F " ( `' F " a ) ) ) )
3229, 31syl5ibrcom 222 . . 3  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( b  =  ( `' F "
a )  ->  a  =  ( F "
b ) ) )
33 f1of1 5813 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -1-1-> B )
343, 33syl 16 . . . . . 6  |-  ( ph  ->  F : A -1-1-> B
)
35 elpwi 4019 . . . . . . 7  |-  ( b  e.  ~P A  -> 
b  C_  A )
3635adantr 465 . . . . . 6  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  b  C_  A )
37 f1imacnv 5830 . . . . . 6  |-  ( ( F : A -1-1-> B  /\  b  C_  A )  ->  ( `' F " ( F " b
) )  =  b )
3834, 36, 37syl2an 477 . . . . 5  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( `' F " ( F " b
) )  =  b )
3938eqcomd 2475 . . . 4  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  b  =  ( `' F " ( F
" b ) ) )
40 imaeq2 5331 . . . . 5  |-  ( a  =  ( F "
b )  ->  ( `' F " a )  =  ( `' F " ( F " b
) ) )
4140eqeq2d 2481 . . . 4  |-  ( a  =  ( F "
b )  ->  (
b  =  ( `' F " a )  <-> 
b  =  ( `' F " ( F
" b ) ) ) )
4239, 41syl5ibrcom 222 . . 3  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( a  =  ( F " b
)  ->  b  =  ( `' F " a ) ) )
4332, 42impbid 191 . 2  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( b  =  ( `' F "
a )  <->  a  =  ( F " b ) ) )
441, 13, 24, 43f1o2d 6509 1  |-  ( ph  ->  ( b  e.  ~P A  |->  ( F "
b ) ) : ~P A -1-1-onto-> ~P B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   ~Pcpw 4010    |-> cmpt 4505   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   -1-1->wf1 5583   -onto->wfo 5584   -1-1-onto->wf1o 5585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593
This theorem is referenced by:  f1opw  6511
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