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Theorem fopwdom 6246
Description: Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
fopwdom  F  _V  F : -onto->  ~P  ~<_  ~P

Proof of Theorem fopwdom
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imassrn 4622 . . . . . 6  `' F " a 
C_  ran  `' F
2 dfdm4 4470 . . . . . . 7  dom  F  ran  `' F
3 fof 5049 . . . . . . . 8  F : -onto->  F : -->
4 fdm 4993 . . . . . . . 8  F : -->  dom 
F
53, 4syl 14 . . . . . . 7  F : -onto->  dom 
F
62, 5syl5eqr 2083 . . . . . 6  F : -onto->  ran  `' F
71, 6syl5sseq 2987 . . . . 5  F : -onto->  `' F " a 
C_
87adantl 262 . . . 4  F  _V  F : -onto->  `' F "
a  C_
9 cnvexg 4798 . . . . . 6  F  _V  `' F  _V
109adantr 261 . . . . 5  F  _V  F : -onto->  `' F  _V
11 imaexg 4623 . . . . 5  `' F  _V  `' F " a  _V
12 elpwg 3359 . . . . 5  `' F " a  _V  `' F " a  ~P  `' F " a 
C_
1310, 11, 123syl 17 . . . 4  F  _V  F : -onto->  `' F " a  ~P  `' F "
a  C_
148, 13mpbird 156 . . 3  F  _V  F : -onto->  `' F "
a  ~P
1514a1d 22 . 2  F  _V  F : -onto->  a  ~P  `' F " a  ~P
16 imaeq2 4607 . . . . . . 7  `' F " a  `' F " b  F " `' F " a  F " `' F " b
1716adantl 262 . . . . . 6  F 
_V  F : -onto-> 
a  ~P  b  ~P  `' F " a  `' F " b  F " `' F " a  F
" `' F " b
18 simpllr 486 . . . . . . 7  F 
_V  F : -onto-> 
a  ~P  b  ~P  `' F " a  `' F " b 
F : -onto->
19 simplrl 487 . . . . . . . 8  F 
_V  F : -onto-> 
a  ~P  b  ~P  `' F " a  `' F " b  a  ~P
2019elpwid 3361 . . . . . . 7  F 
_V  F : -onto-> 
a  ~P  b  ~P  `' F " a  `' F " b  a  C_
21 foimacnv 5087 . . . . . . 7  F : -onto->  a  C_  F " `' F " a  a
2218, 20, 21syl2anc 391 . . . . . 6  F 
_V  F : -onto-> 
a  ~P  b  ~P  `' F " a  `' F " b  F " `' F " a  a
23 simplrr 488 . . . . . . . 8  F 
_V  F : -onto-> 
a  ~P  b  ~P  `' F " a  `' F " b  b  ~P
2423elpwid 3361 . . . . . . 7  F 
_V  F : -onto-> 
a  ~P  b  ~P  `' F " a  `' F " b  b  C_
25 foimacnv 5087 . . . . . . 7  F : -onto->  b  C_  F " `' F " b  b
2618, 24, 25syl2anc 391 . . . . . 6  F 
_V  F : -onto-> 
a  ~P  b  ~P  `' F " a  `' F " b  F " `' F " b  b
2717, 22, 263eqtr3d 2077 . . . . 5  F 
_V  F : -onto-> 
a  ~P  b  ~P  `' F " a  `' F " b  a  b
2827ex 108 . . . 4  F  _V  F : -onto->  a 
~P  b  ~P  `' F " a  `' F " b  a  b
29 imaeq2 4607 . . . 4  a  b  `' F " a  `' F " b
3028, 29impbid1 130 . . 3  F  _V  F : -onto->  a 
~P  b  ~P  `' F " a  `' F " b  a  b
3130ex 108 . 2  F  _V  F : -onto->  a 
~P  b  ~P  `' F "
a  `' F " b  a  b
32 rnexg 4540 . . . . 5  F  _V  ran  F  _V
33 forn 5052 . . . . . 6  F : -onto->  ran 
F
3433eleq1d 2103 . . . . 5  F : -onto->  ran  F  _V  _V
3532, 34syl5ibcom 144 . . . 4  F  _V  F : -onto->  _V
3635imp 115 . . 3  F  _V  F : -onto->  _V
37 pwexg 3924 . . 3  _V  ~P  _V
3836, 37syl 14 . 2  F  _V  F : -onto->  ~P  _V
39 dmfex 5022 . . . 4  F  _V  F : -->  _V
403, 39sylan2 270 . . 3  F  _V  F : -onto->  _V
41 pwexg 3924 . . 3  _V  ~P  _V
4240, 41syl 14 . 2  F  _V  F : -onto->  ~P  _V
4315, 31, 38, 42dom3d 6190 1  F  _V  F : -onto->  ~P  ~<_  ~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   class class class wbr 3755   `'ccnv 4287   dom cdm 4288   ran crn 4289   "cima 4291   -->wf 4841   -onto->wfo 4843    ~<_ cdom 6156
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-13 1401  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  ax-un 4136
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-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  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-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-fv 4853  df-dom 6159
This theorem is referenced by: (None)
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