MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1opw2 Structured version   Unicode version

Theorem f1opw2 6313
Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6314 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 2443 . 2  |-  ( b  e.  ~P A  |->  ( F " b ) )  =  ( b  e.  ~P A  |->  ( F " b ) )
2 imassrn 5180 . . . . 5  |-  ( F
" b )  C_  ran  F
3 f1opw2.1 . . . . . . 7  |-  ( ph  ->  F : A -1-1-onto-> B )
4 f1ofo 5648 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
53, 4syl 16 . . . . . 6  |-  ( ph  ->  F : A -onto-> B
)
6 forn 5623 . . . . . 6  |-  ( F : A -onto-> B  ->  ran  F  =  B )
75, 6syl 16 . . . . 5  |-  ( ph  ->  ran  F  =  B )
82, 7syl5sseq 3404 . . . 4  |-  ( ph  ->  ( F " b
)  C_  B )
9 f1opw2.3 . . . . 5  |-  ( ph  ->  ( F " b
)  e.  _V )
10 elpwg 3868 . . . . 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 5180 . . . . 5  |-  ( `' F " a ) 
C_  ran  `' F
15 dfdm4 5032 . . . . . 6  |-  dom  F  =  ran  `' F
16 f1odm 5645 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  dom  F  =  A )
173, 16syl 16 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
1815, 17syl5eqr 2489 . . . . 5  |-  ( ph  ->  ran  `' F  =  A )
1914, 18syl5sseq 3404 . . . 4  |-  ( ph  ->  ( `' F "
a )  C_  A
)
20 f1opw2.2 . . . . 5  |-  ( ph  ->  ( `' F "
a )  e.  _V )
21 elpwg 3868 . . . . 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 3869 . . . . . . 7  |-  ( a  e.  ~P B  -> 
a  C_  B )
2625adantl 466 . . . . . 6  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  a  C_  B )
27 foimacnv 5658 . . . . . 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 2448 . . . 4  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  a  =  ( F " ( `' F " a ) ) )
30 imaeq2 5165 . . . . 5  |-  ( b  =  ( `' F " a )  ->  ( F " b )  =  ( F " ( `' F " a ) ) )
3130eqeq2d 2454 . . . 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 5640 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -1-1-> B )
343, 33syl 16 . . . . . 6  |-  ( ph  ->  F : A -1-1-> B
)
35 elpwi 3869 . . . . . . 7  |-  ( b  e.  ~P A  -> 
b  C_  A )
3635adantr 465 . . . . . 6  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  b  C_  A )
37 f1imacnv 5657 . . . . . 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 2448 . . . 4  |-  ( (
ph  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  ->  b  =  ( `' F " ( F
" b ) ) )
40 imaeq2 5165 . . . . 5  |-  ( a  =  ( F "
b )  ->  ( `' F " a )  =  ( `' F " ( F " b
) ) )
4140eqeq2d 2454 . . . 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 6312 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 1369    e. wcel 1756   _Vcvv 2972    C_ wss 3328   ~Pcpw 3860    e. cmpt 4350   `'ccnv 4839   dom cdm 4840   ran crn 4841   "cima 4843   -1-1->wf1 5415   -onto->wfo 5416   -1-1-onto->wf1o 5417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425
This theorem is referenced by:  f1opw  6314
  Copyright terms: Public domain W3C validator