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Theorem fopwdom 7625
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 5348 . . . . . 6  |-  ( `' F " a ) 
C_  ran  `' F
2 dfdm4 5195 . . . . . . 7  |-  dom  F  =  ran  `' F
3 fof 5795 . . . . . . . 8  |-  ( F : A -onto-> B  ->  F : A --> B )
4 fdm 5735 . . . . . . . 8  |-  ( F : A --> B  ->  dom  F  =  A )
53, 4syl 16 . . . . . . 7  |-  ( F : A -onto-> B  ->  dom  F  =  A )
62, 5syl5eqr 2522 . . . . . 6  |-  ( F : A -onto-> B  ->  ran  `' F  =  A
)
71, 6syl5sseq 3552 . . . . 5  |-  ( F : A -onto-> B  -> 
( `' F "
a )  C_  A
)
87adantl 466 . . . 4  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( `' F " a )  C_  A
)
9 cnvexg 6730 . . . . . 6  |-  ( F  e.  _V  ->  `' F  e.  _V )
109adantr 465 . . . . 5  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  `' F  e. 
_V )
11 imaexg 6721 . . . . 5  |-  ( `' F  e.  _V  ->  ( `' F " a )  e.  _V )
12 elpwg 4018 . . . . 5  |-  ( ( `' F " a )  e.  _V  ->  (
( `' F "
a )  e.  ~P A 
<->  ( `' F "
a )  C_  A
) )
1310, 11, 123syl 20 . . . 4  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( ( `' F " a )  e.  ~P A  <->  ( `' F " a )  C_  A ) )
148, 13mpbird 232 . . 3  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( `' F " a )  e.  ~P A )
1514a1d 25 . 2  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( a  e. 
~P B  ->  ( `' F " a )  e.  ~P A ) )
16 imaeq2 5333 . . . . . . 7  |-  ( ( `' F " a )  =  ( `' F " b )  ->  ( F " ( `' F " a ) )  =  ( F " ( `' F " b ) ) )
1716adantl 466 . . . . . 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 758 . . . . . . 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 759 . . . . . . . 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 4020 . . . . . . 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 5833 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  a  C_  B )  ->  ( F "
( `' F "
a ) )  =  a )
2218, 20, 21syl2anc 661 . . . . . 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 760 . . . . . . . 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 4020 . . . . . . 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 5833 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  b  C_  B )  ->  ( F "
( `' F "
b ) )  =  b )
2618, 24, 25syl2anc 661 . . . . . 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 2516 . . . . 5  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
a  =  b )
2827ex 434 . . . 4  |-  ( ( ( F  e.  _V  /\  F : A -onto-> B
)  /\  ( a  e.  ~P B  /\  b  e.  ~P B ) )  ->  ( ( `' F " a )  =  ( `' F " b )  ->  a  =  b ) )
29 imaeq2 5333 . . . 4  |-  ( a  =  b  ->  ( `' F " a )  =  ( `' F " b ) )
3028, 29impbid1 203 . . 3  |-  ( ( ( F  e.  _V  /\  F : A -onto-> B
)  /\  ( a  e.  ~P B  /\  b  e.  ~P B ) )  ->  ( ( `' F " a )  =  ( `' F " b )  <->  a  =  b ) )
3130ex 434 . 2  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( ( a  e.  ~P B  /\  b  e.  ~P B
)  ->  ( ( `' F " a )  =  ( `' F " b )  <->  a  =  b ) ) )
32 rnexg 6716 . . . . 5  |-  ( F  e.  _V  ->  ran  F  e.  _V )
33 forn 5798 . . . . . 6  |-  ( F : A -onto-> B  ->  ran  F  =  B )
3433eleq1d 2536 . . . . 5  |-  ( F : A -onto-> B  -> 
( ran  F  e.  _V 
<->  B  e.  _V )
)
3532, 34syl5ibcom 220 . . . 4  |-  ( F  e.  _V  ->  ( F : A -onto-> B  ->  B  e.  _V )
)
3635imp 429 . . 3  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  B  e.  _V )
37 pwexg 4631 . . 3  |-  ( B  e.  _V  ->  ~P B  e.  _V )
3836, 37syl 16 . 2  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ~P B  e. 
_V )
39 dmfex 6742 . . . 4  |-  ( ( F  e.  _V  /\  F : A --> B )  ->  A  e.  _V )
403, 39sylan2 474 . . 3  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  A  e.  _V )
41 pwexg 4631 . . 3  |-  ( A  e.  _V  ->  ~P A  e.  _V )
4240, 41syl 16 . 2  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ~P A  e. 
_V )
4315, 31, 38, 42dom3d 7557 1  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ~P B  ~<_  ~P A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   ~Pcpw 4010   class class class wbr 4447   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   -->wf 5584   -onto->wfo 5586    ~<_ cdom 7514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-fv 5596  df-dom 7518
This theorem is referenced by:  pwdom  7669  wdompwdom  8004
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