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Theorem f1opw2 6509
Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6510 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 2402 . 2  |-  ( b  e.  ~P A  |->  ( F " b ) )  =  ( b  e.  ~P A  |->  ( F " b ) )
2 imassrn 5168 . . . . 5  |-  ( F
" b )  C_  ran  F
3 f1opw2.1 . . . . . . 7  |-  ( ph  ->  F : A -1-1-onto-> B )
4 f1ofo 5806 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
53, 4syl 17 . . . . . 6  |-  ( ph  ->  F : A -onto-> B
)
6 forn 5781 . . . . . 6  |-  ( F : A -onto-> B  ->  ran  F  =  B )
75, 6syl 17 . . . . 5  |-  ( ph  ->  ran  F  =  B )
82, 7syl5sseq 3490 . . . 4  |-  ( ph  ->  ( F " b
)  C_  B )
9 f1opw2.3 . . . . 5  |-  ( ph  ->  ( F " b
)  e.  _V )
10 elpwg 3963 . . . . 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 232 . . 3  |-  ( ph  ->  ( F " b
)  e.  ~P B
)
1312adantr 463 . 2  |-  ( (
ph  /\  b  e.  ~P A )  ->  ( F " b )  e. 
~P B )
14 imassrn 5168 . . . . 5  |-  ( `' F " a ) 
C_  ran  `' F
15 dfdm4 5016 . . . . . 6  |-  dom  F  =  ran  `' F
16 f1odm 5803 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  dom  F  =  A )
173, 16syl 17 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
1815, 17syl5eqr 2457 . . . . 5  |-  ( ph  ->  ran  `' F  =  A )
1914, 18syl5sseq 3490 . . . 4  |-  ( ph  ->  ( `' F "
a )  C_  A
)
20 f1opw2.2 . . . . 5  |-  ( ph  ->  ( `' F "
a )  e.  _V )
21 elpwg 3963 . . . . 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 232 . . 3  |-  ( ph  ->  ( `' F "
a )  e.  ~P A )
2423adantr 463 . 2  |-  ( (
ph  /\  a  e.  ~P B )  ->  ( `' F " a )  e.  ~P A )
25 elpwi 3964 . . . . . . 7  |-  ( a  e.  ~P B  -> 
a  C_  B )
2625adantl 464 . . . . . 6  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  a  C_  B )
27 foimacnv 5816 . . . . . 6  |-  ( ( F : A -onto-> B  /\  a  C_  B )  ->  ( F "
( `' F "
a ) )  =  a )
285, 26, 27syl2an 475 . . . . 5  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( F "
( `' F "
a ) )  =  a )
2928eqcomd 2410 . . . 4  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  a  =  ( F " ( `' F " a ) ) )
30 imaeq2 5153 . . . . 5  |-  ( b  =  ( `' F " a )  ->  ( F " b )  =  ( F " ( `' F " a ) ) )
3130eqeq2d 2416 . . . 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 5798 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -1-1-> B )
343, 33syl 17 . . . . . 6  |-  ( ph  ->  F : A -1-1-> B
)
35 elpwi 3964 . . . . . . 7  |-  ( b  e.  ~P A  -> 
b  C_  A )
3635adantr 463 . . . . . 6  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  b  C_  A )
37 f1imacnv 5815 . . . . . 6  |-  ( ( F : A -1-1-> B  /\  b  C_  A )  ->  ( `' F " ( F " b
) )  =  b )
3834, 36, 37syl2an 475 . . . . 5  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( `' F " ( F " b
) )  =  b )
3938eqcomd 2410 . . . 4  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  b  =  ( `' F " ( F
" b ) ) )
40 imaeq2 5153 . . . . 5  |-  ( a  =  ( F "
b )  ->  ( `' F " a )  =  ( `' F " ( F " b
) ) )
4140eqeq2d 2416 . . . 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 190 . 2  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( b  =  ( `' F "
a )  <->  a  =  ( F " b ) ) )
441, 13, 24, 43f1o2d 6508 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 367    = wceq 1405    e. wcel 1842   _Vcvv 3059    C_ wss 3414   ~Pcpw 3955    |-> cmpt 4453   `'ccnv 4822   dom cdm 4823   ran crn 4824   "cima 4826   -1-1->wf1 5566   -onto->wfo 5567   -1-1-onto->wf1o 5568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576
This theorem is referenced by:  f1opw  6510
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