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Theorem fopwdom 6681
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  e.  _V  /\  F : A -onto-> B )  ->  ~P B  ~<_  ~P A )

Proof of Theorem fopwdom
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imassrn 4848 . . . . . 6  |-  ( `' F " a ) 
C_  ran  `' F
2 dfdm4 4689 . . . . . . 7  |-  dom  F  =  ran  `' F
3 fof 5301 . . . . . . . 8  |-  ( F : A -onto-> B  ->  F : A --> B )
4 fdm 5234 . . . . . . . 8  |-  ( F : A --> B  ->  dom  F  =  A )
53, 4syl 14 . . . . . . 7  |-  ( F : A -onto-> B  ->  dom  F  =  A )
62, 5syl5eqr 2159 . . . . . 6  |-  ( F : A -onto-> B  ->  ran  `' F  =  A
)
71, 6sseqtrid 3111 . . . . 5  |-  ( F : A -onto-> B  -> 
( `' F "
a )  C_  A
)
87adantl 273 . . . 4  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( `' F " a )  C_  A
)
9 cnvexg 5032 . . . . . 6  |-  ( F  e.  _V  ->  `' F  e.  _V )
109adantr 272 . . . . 5  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  `' F  e. 
_V )
11 imaexg 4849 . . . . 5  |-  ( `' F  e.  _V  ->  ( `' F " a )  e.  _V )
12 elpwg 3482 . . . . 5  |-  ( ( `' F " a )  e.  _V  ->  (
( `' F "
a )  e.  ~P A 
<->  ( `' F "
a )  C_  A
) )
1310, 11, 123syl 17 . . . 4  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( ( `' F " a )  e.  ~P A  <->  ( `' F " a )  C_  A ) )
148, 13mpbird 166 . . 3  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( `' F " a )  e.  ~P A )
1514a1d 22 . 2  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( a  e. 
~P B  ->  ( `' F " a )  e.  ~P A ) )
16 imaeq2 4833 . . . . . . 7  |-  ( ( `' F " a )  =  ( `' F " b )  ->  ( F " ( `' F " a ) )  =  ( F " ( `' F " b ) ) )
1716adantl 273 . . . . . 6  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
( F " ( `' F " a ) )  =  ( F
" ( `' F " b ) ) )
18 simpllr 506 . . . . . . 7  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  ->  F : A -onto-> B )
19 simplrl 507 . . . . . . . 8  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
a  e.  ~P B
)
2019elpwid 3485 . . . . . . 7  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
a  C_  B )
21 foimacnv 5339 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  a  C_  B )  ->  ( F "
( `' F "
a ) )  =  a )
2218, 20, 21syl2anc 406 . . . . . 6  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
( F " ( `' F " a ) )  =  a )
23 simplrr 508 . . . . . . . 8  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
b  e.  ~P B
)
2423elpwid 3485 . . . . . . 7  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
b  C_  B )
25 foimacnv 5339 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  b  C_  B )  ->  ( F "
( `' F "
b ) )  =  b )
2618, 24, 25syl2anc 406 . . . . . 6  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
( F " ( `' F " b ) )  =  b )
2717, 22, 263eqtr3d 2153 . . . . 5  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
a  =  b )
2827ex 114 . . . 4  |-  ( ( ( F  e.  _V  /\  F : A -onto-> B
)  /\  ( a  e.  ~P B  /\  b  e.  ~P B ) )  ->  ( ( `' F " a )  =  ( `' F " b )  ->  a  =  b ) )
29 imaeq2 4833 . . . 4  |-  ( a  =  b  ->  ( `' F " a )  =  ( `' F " b ) )
3028, 29impbid1 141 . . 3  |-  ( ( ( F  e.  _V  /\  F : A -onto-> B
)  /\  ( a  e.  ~P B  /\  b  e.  ~P B ) )  ->  ( ( `' F " a )  =  ( `' F " b )  <->  a  =  b ) )
3130ex 114 . 2  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( ( a  e.  ~P B  /\  b  e.  ~P B
)  ->  ( ( `' F " a )  =  ( `' F " b )  <->  a  =  b ) ) )
32 rnexg 4760 . . . . 5  |-  ( F  e.  _V  ->  ran  F  e.  _V )
33 forn 5304 . . . . . 6  |-  ( F : A -onto-> B  ->  ran  F  =  B )
3433eleq1d 2181 . . . . 5  |-  ( F : A -onto-> B  -> 
( ran  F  e.  _V 
<->  B  e.  _V )
)
3532, 34syl5ibcom 154 . . . 4  |-  ( F  e.  _V  ->  ( F : A -onto-> B  ->  B  e.  _V )
)
3635imp 123 . . 3  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  B  e.  _V )
37 pwexg 4062 . . 3  |-  ( B  e.  _V  ->  ~P B  e.  _V )
3836, 37syl 14 . 2  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ~P B  e. 
_V )
39 dmfex 5268 . . . 4  |-  ( ( F  e.  _V  /\  F : A --> B )  ->  A  e.  _V )
403, 39sylan2 282 . . 3  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  A  e.  _V )
41 pwexg 4062 . . 3  |-  ( A  e.  _V  ->  ~P A  e.  _V )
4240, 41syl 14 . 2  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ~P A  e. 
_V )
4315, 31, 38, 42dom3d 6620 1  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ~P B  ~<_  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1312    e. wcel 1461   _Vcvv 2655    C_ wss 3035   ~Pcpw 3474   class class class wbr 3893   `'ccnv 4496   dom cdm 4497   ran crn 4498   "cima 4500   -->wf 5075   -onto->wfo 5077    ~<_ cdom 6585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-fv 5087  df-dom 6588
This theorem is referenced by: (None)
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