ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1opw2 Unicode version

Theorem f1opw2 5734
Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 5735 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 2056 . 2  |-  ( b  e.  ~P A  |->  ( F " b ) )  =  ( b  e.  ~P A  |->  ( F " b ) )
2 imassrn 4707 . . . . 5  |-  ( F
" b )  C_  ran  F
3 f1opw2.1 . . . . . . 7  |-  ( ph  ->  F : A -1-1-onto-> B )
4 f1ofo 5161 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
53, 4syl 14 . . . . . 6  |-  ( ph  ->  F : A -onto-> B
)
6 forn 5137 . . . . . 6  |-  ( F : A -onto-> B  ->  ran  F  =  B )
75, 6syl 14 . . . . 5  |-  ( ph  ->  ran  F  =  B )
82, 7syl5sseq 3021 . . . 4  |-  ( ph  ->  ( F " b
)  C_  B )
9 f1opw2.3 . . . . 5  |-  ( ph  ->  ( F " b
)  e.  _V )
10 elpwg 3395 . . . . 5  |-  ( ( F " b )  e.  _V  ->  (
( F " b
)  e.  ~P B  <->  ( F " b ) 
C_  B ) )
119, 10syl 14 . . . 4  |-  ( ph  ->  ( ( F "
b )  e.  ~P B 
<->  ( F " b
)  C_  B )
)
128, 11mpbird 160 . . 3  |-  ( ph  ->  ( F " b
)  e.  ~P B
)
1312adantr 265 . 2  |-  ( (
ph  /\  b  e.  ~P A )  ->  ( F " b )  e. 
~P B )
14 imassrn 4707 . . . . 5  |-  ( `' F " a ) 
C_  ran  `' F
15 dfdm4 4555 . . . . . 6  |-  dom  F  =  ran  `' F
16 f1odm 5158 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  dom  F  =  A )
173, 16syl 14 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
1815, 17syl5eqr 2102 . . . . 5  |-  ( ph  ->  ran  `' F  =  A )
1914, 18syl5sseq 3021 . . . 4  |-  ( ph  ->  ( `' F "
a )  C_  A
)
20 f1opw2.2 . . . . 5  |-  ( ph  ->  ( `' F "
a )  e.  _V )
21 elpwg 3395 . . . . 5  |-  ( ( `' F " a )  e.  _V  ->  (
( `' F "
a )  e.  ~P A 
<->  ( `' F "
a )  C_  A
) )
2220, 21syl 14 . . . 4  |-  ( ph  ->  ( ( `' F " a )  e.  ~P A 
<->  ( `' F "
a )  C_  A
) )
2319, 22mpbird 160 . . 3  |-  ( ph  ->  ( `' F "
a )  e.  ~P A )
2423adantr 265 . 2  |-  ( (
ph  /\  a  e.  ~P B )  ->  ( `' F " a )  e.  ~P A )
25 elpwi 3396 . . . . . . 7  |-  ( a  e.  ~P B  -> 
a  C_  B )
2625adantl 266 . . . . . 6  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  a  C_  B )
27 foimacnv 5172 . . . . . 6  |-  ( ( F : A -onto-> B  /\  a  C_  B )  ->  ( F "
( `' F "
a ) )  =  a )
285, 26, 27syl2an 277 . . . . 5  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( F "
( `' F "
a ) )  =  a )
2928eqcomd 2061 . . . 4  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  a  =  ( F " ( `' F " a ) ) )
30 imaeq2 4692 . . . . 5  |-  ( b  =  ( `' F " a )  ->  ( F " b )  =  ( F " ( `' F " a ) ) )
3130eqeq2d 2067 . . . 4  |-  ( b  =  ( `' F " a )  ->  (
a  =  ( F
" b )  <->  a  =  ( F " ( `' F " a ) ) ) )
3229, 31syl5ibrcom 150 . . 3  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( b  =  ( `' F "
a )  ->  a  =  ( F "
b ) ) )
33 f1of1 5153 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -1-1-> B )
343, 33syl 14 . . . . . 6  |-  ( ph  ->  F : A -1-1-> B
)
35 elpwi 3396 . . . . . . 7  |-  ( b  e.  ~P A  -> 
b  C_  A )
3635adantr 265 . . . . . 6  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  b  C_  A )
37 f1imacnv 5171 . . . . . 6  |-  ( ( F : A -1-1-> B  /\  b  C_  A )  ->  ( `' F " ( F " b
) )  =  b )
3834, 36, 37syl2an 277 . . . . 5  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( `' F " ( F " b
) )  =  b )
3938eqcomd 2061 . . . 4  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  b  =  ( `' F " ( F
" b ) ) )
40 imaeq2 4692 . . . . 5  |-  ( a  =  ( F "
b )  ->  ( `' F " a )  =  ( `' F " ( F " b
) ) )
4140eqeq2d 2067 . . . 4  |-  ( a  =  ( F "
b )  ->  (
b  =  ( `' F " a )  <-> 
b  =  ( `' F " ( F
" b ) ) ) )
4239, 41syl5ibrcom 150 . . 3  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( a  =  ( F " b
)  ->  b  =  ( `' F " a ) ) )
4332, 42impbid 124 . 2  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  ( b  =  ( `' F "
a )  <->  a  =  ( F " b ) ) )
441, 13, 24, 43f1o2d 5733 1  |-  ( ph  ->  ( b  e.  ~P A  |->  ( F "
b ) ) : ~P A -1-1-onto-> ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   _Vcvv 2574    C_ wss 2945   ~Pcpw 3387    |-> cmpt 3846   `'ccnv 4372   dom cdm 4373   ran crn 4374   "cima 4376   -1-1->wf1 4927   -onto->wfo 4928   -1-1-onto->wf1o 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937
This theorem is referenced by:  f1opw  5735
  Copyright terms: Public domain W3C validator