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Theorem f1opw2 5648
Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 5649 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
f1opw2.1  F : -1-1-onto->
f1opw2.2  `' F "
a  _V
f1opw2.3  F " b  _V
Assertion
Ref Expression
f1opw2  b  ~P  |->  F "
b : ~P -1-1-onto-> ~P
Distinct variable groups:    a, b,   , a, b    F, a, b   , a, b

Proof of Theorem f1opw2
StepHypRef Expression
1 eqid 2037 . 2  b  ~P  |->  F " b  b  ~P  |->  F " b
2 imassrn 4622 . . . . 5  F
" b  C_  ran  F
3 f1opw2.1 . . . . . . 7  F : -1-1-onto->
4 f1ofo 5076 . . . . . . 7  F : -1-1-onto->  F : -onto->
53, 4syl 14 . . . . . 6  F : -onto->
6 forn 5052 . . . . . 6  F : -onto->  ran 
F
75, 6syl 14 . . . . 5  ran  F
82, 7syl5sseq 2987 . . . 4  F " b  C_
9 f1opw2.3 . . . . 5  F " b  _V
10 elpwg 3359 . . . . 5  F " b  _V  F " b  ~P  F
" b  C_
119, 10syl 14 . . . 4  F "
b  ~P  F " b  C_
128, 11mpbird 156 . . 3  F " b  ~P
1312adantr 261 . 2  b  ~P  F " b 
~P
14 imassrn 4622 . . . . 5  `' F " a 
C_  ran  `' F
15 dfdm4 4470 . . . . . 6  dom  F  ran  `' F
16 f1odm 5073 . . . . . . 7  F : -1-1-onto->  dom  F
173, 16syl 14 . . . . . 6  dom  F
1815, 17syl5eqr 2083 . . . . 5  ran  `' F
1914, 18syl5sseq 2987 . . . 4  `' F "
a  C_
20 f1opw2.2 . . . . 5  `' F "
a  _V
21 elpwg 3359 . . . . 5  `' F " a  _V  `' F " a  ~P  `' F " a 
C_
2220, 21syl 14 . . . 4  `' F " a  ~P  `' F "
a  C_
2319, 22mpbird 156 . . 3  `' F "
a  ~P
2423adantr 261 . 2  a  ~P  `' F " a  ~P
25 elpwi 3360 . . . . . . 7  a  ~P  a  C_
2625adantl 262 . . . . . 6  b  ~P  a  ~P  a  C_
27 foimacnv 5087 . . . . . 6  F : -onto->  a  C_  F " `' F " a  a
285, 26, 27syl2an 273 . . . . 5  b 
~P  a  ~P  F " `' F " a  a
2928eqcomd 2042 . . . 4  b 
~P  a  ~P  a  F
" `' F " a
30 imaeq2 4607 . . . . 5  b  `' F " a  F " b  F " `' F " a
3130eqeq2d 2048 . . . 4  b  `' F " a 
a  F
" b  a  F " `' F " a
3229, 31syl5ibrcom 146 . . 3  b 
~P  a  ~P  b  `' F " a  a  F
" b
33 f1of1 5068 . . . . . . 7  F : -1-1-onto->  F : -1-1->
343, 33syl 14 . . . . . 6  F : -1-1->
35 elpwi 3360 . . . . . . 7  b  ~P  b  C_
3635adantr 261 . . . . . 6  b  ~P  a  ~P  b  C_
37 f1imacnv 5086 . . . . . 6  F : -1-1->  b  C_  `' F " F " b  b
3834, 36, 37syl2an 273 . . . . 5  b 
~P  a  ~P  `' F " F " b  b
3938eqcomd 2042 . . . 4  b 
~P  a  ~P  b  `' F " F
" b
40 imaeq2 4607 . . . . 5  a  F "
b  `' F " a  `' F " F " b
4140eqeq2d 2048 . . . 4  a  F "
b 
b  `' F " a  b  `' F " F
" b
4239, 41syl5ibrcom 146 . . 3  b 
~P  a  ~P  a  F " b  b  `' F " a
4332, 42impbid 120 . 2  b 
~P  a  ~P  b  `' F " a  a  F
" b
441, 13, 24, 43f1o2d 5647 1  b  ~P  |->  F "
b : ~P -1-1-onto-> ~P
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390   _Vcvv 2551    C_ wss 2911   ~Pcpw 3351    |-> cmpt 3809   `'ccnv 4287   dom cdm 4288   ran crn 4289   "cima 4291   -1-1->wf1 4842   -onto->wfo 4843   -1-1-onto->wf1o 4844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by:  f1opw  5649
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