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Theorem f1opw2 6519
Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6520 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 2450 . 2  |-  ( b  e.  ~P A  |->  ( F " b ) )  =  ( b  e.  ~P A  |->  ( F " b ) )
2 imassrn 5178 . . . . 5  |-  ( F
" b )  C_  ran  F
3 f1opw2.1 . . . . . . 7  |-  ( ph  ->  F : A -1-1-onto-> B )
4 f1ofo 5819 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
53, 4syl 17 . . . . . 6  |-  ( ph  ->  F : A -onto-> B
)
6 forn 5794 . . . . . 6  |-  ( F : A -onto-> B  ->  ran  F  =  B )
75, 6syl 17 . . . . 5  |-  ( ph  ->  ran  F  =  B )
82, 7syl5sseq 3479 . . . 4  |-  ( ph  ->  ( F " b
)  C_  B )
9 f1opw2.3 . . . . 5  |-  ( ph  ->  ( F " b
)  e.  _V )
10 elpwg 3958 . . . . 5  |-  ( ( F " b )  e.  _V  ->  (
( F " b
)  e.  ~P B  <->  ( F " b ) 
C_  B ) )
119, 10syl 17 . . . 4  |-  ( ph  ->  ( ( F "
b )  e.  ~P B 
<->  ( F " b
)  C_  B )
)
128, 11mpbird 236 . . 3  |-  ( ph  ->  ( F " b
)  e.  ~P B
)
1312adantr 467 . 2  |-  ( (
ph  /\  b  e.  ~P A )  ->  ( F " b )  e. 
~P B )
14 imassrn 5178 . . . . 5  |-  ( `' F " a ) 
C_  ran  `' F
15 dfdm4 5026 . . . . . 6  |-  dom  F  =  ran  `' F
16 f1odm 5816 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  dom  F  =  A )
173, 16syl 17 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
1815, 17syl5eqr 2498 . . . . 5  |-  ( ph  ->  ran  `' F  =  A )
1914, 18syl5sseq 3479 . . . 4  |-  ( ph  ->  ( `' F "
a )  C_  A
)
20 f1opw2.2 . . . . 5  |-  ( ph  ->  ( `' F "
a )  e.  _V )
21 elpwg 3958 . . . . 5  |-  ( ( `' F " a )  e.  _V  ->  (
( `' F "
a )  e.  ~P A 
<->  ( `' F "
a )  C_  A
) )
2220, 21syl 17 . . . 4  |-  ( ph  ->  ( ( `' F " a )  e.  ~P A 
<->  ( `' F "
a )  C_  A
) )
2319, 22mpbird 236 . . 3  |-  ( ph  ->  ( `' F "
a )  e.  ~P A )
2423adantr 467 . 2  |-  ( (
ph  /\  a  e.  ~P B )  ->  ( `' F " a )  e.  ~P A )
25 elpwi 3959 . . . . . . 7  |-  ( a  e.  ~P B  -> 
a  C_  B )
2625adantl 468 . . . . . 6  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  a  C_  B )
27 foimacnv 5829 . . . . . 6  |-  ( ( F : A -onto-> B  /\  a  C_  B )  ->  ( F "
( `' F "
a ) )  =  a )
285, 26, 27syl2an 480 . . . . 5  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( F "
( `' F "
a ) )  =  a )
2928eqcomd 2456 . . . 4  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  a  =  ( F " ( `' F " a ) ) )
30 imaeq2 5163 . . . . 5  |-  ( b  =  ( `' F " a )  ->  ( F " b )  =  ( F " ( `' F " a ) ) )
3130eqeq2d 2460 . . . 4  |-  ( b  =  ( `' F " a )  ->  (
a  =  ( F
" b )  <->  a  =  ( F " ( `' F " a ) ) ) )
3229, 31syl5ibrcom 226 . . 3  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( b  =  ( `' F "
a )  ->  a  =  ( F "
b ) ) )
33 f1of1 5811 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -1-1-> B )
343, 33syl 17 . . . . . 6  |-  ( ph  ->  F : A -1-1-> B
)
35 elpwi 3959 . . . . . . 7  |-  ( b  e.  ~P A  -> 
b  C_  A )
3635adantr 467 . . . . . 6  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  b  C_  A )
37 f1imacnv 5828 . . . . . 6  |-  ( ( F : A -1-1-> B  /\  b  C_  A )  ->  ( `' F " ( F " b
) )  =  b )
3834, 36, 37syl2an 480 . . . . 5  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( `' F " ( F " b
) )  =  b )
3938eqcomd 2456 . . . 4  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  b  =  ( `' F " ( F
" b ) ) )
40 imaeq2 5163 . . . . 5  |-  ( a  =  ( F "
b )  ->  ( `' F " a )  =  ( `' F " ( F " b
) ) )
4140eqeq2d 2460 . . . 4  |-  ( a  =  ( F "
b )  ->  (
b  =  ( `' F " a )  <-> 
b  =  ( `' F " ( F
" b ) ) ) )
4239, 41syl5ibrcom 226 . . 3  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( a  =  ( F " b
)  ->  b  =  ( `' F " a ) ) )
4332, 42impbid 194 . 2  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( b  =  ( `' F "
a )  <->  a  =  ( F " b ) ) )
441, 13, 24, 43f1o2d 6518 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 188    /\ wa 371    = wceq 1443    e. wcel 1886   _Vcvv 3044    C_ wss 3403   ~Pcpw 3950    |-> cmpt 4460   `'ccnv 4832   dom cdm 4833   ran crn 4834   "cima 4836   -1-1->wf1 5578   -onto->wfo 5579   -1-1-onto->wf1o 5580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588
This theorem is referenced by:  f1opw  6520
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