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Theorem fopwdom 6532
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 4772 . . . . . 6  |-  ( `' F " a ) 
C_  ran  `' F
2 dfdm4 4616 . . . . . . 7  |-  dom  F  =  ran  `' F
3 fof 5217 . . . . . . . 8  |-  ( F : A -onto-> B  ->  F : A --> B )
4 fdm 5152 . . . . . . . 8  |-  ( F : A --> B  ->  dom  F  =  A )
53, 4syl 14 . . . . . . 7  |-  ( F : A -onto-> B  ->  dom  F  =  A )
62, 5syl5eqr 2134 . . . . . 6  |-  ( F : A -onto-> B  ->  ran  `' F  =  A
)
71, 6syl5sseq 3072 . . . . 5  |-  ( F : A -onto-> B  -> 
( `' F "
a )  C_  A
)
87adantl 271 . . . 4  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( `' F " a )  C_  A
)
9 cnvexg 4955 . . . . . 6  |-  ( F  e.  _V  ->  `' F  e.  _V )
109adantr 270 . . . . 5  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  `' F  e. 
_V )
11 imaexg 4773 . . . . 5  |-  ( `' F  e.  _V  ->  ( `' F " a )  e.  _V )
12 elpwg 3433 . . . . 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 165 . . 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 4757 . . . . . . 7  |-  ( ( `' F " a )  =  ( `' F " b )  ->  ( F " ( `' F " a ) )  =  ( F " ( `' F " b ) ) )
1716adantl 271 . . . . . 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 501 . . . . . . 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 502 . . . . . . . 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 3435 . . . . . . 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 5255 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  a  C_  B )  ->  ( F "
( `' F "
a ) )  =  a )
2218, 20, 21syl2anc 403 . . . . . 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 503 . . . . . . . 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 3435 . . . . . . 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 5255 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  b  C_  B )  ->  ( F "
( `' F "
b ) )  =  b )
2618, 24, 25syl2anc 403 . . . . . 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 2128 . . . . 5  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
a  =  b )
2827ex 113 . . . 4  |-  ( ( ( F  e.  _V  /\  F : A -onto-> B
)  /\  ( a  e.  ~P B  /\  b  e.  ~P B ) )  ->  ( ( `' F " a )  =  ( `' F " b )  ->  a  =  b ) )
29 imaeq2 4757 . . . 4  |-  ( a  =  b  ->  ( `' F " a )  =  ( `' F " b ) )
3028, 29impbid1 140 . . 3  |-  ( ( ( F  e.  _V  /\  F : A -onto-> B
)  /\  ( a  e.  ~P B  /\  b  e.  ~P B ) )  ->  ( ( `' F " a )  =  ( `' F " b )  <->  a  =  b ) )
3130ex 113 . 2  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( ( a  e.  ~P B  /\  b  e.  ~P B
)  ->  ( ( `' F " a )  =  ( `' F " b )  <->  a  =  b ) ) )
32 rnexg 4686 . . . . 5  |-  ( F  e.  _V  ->  ran  F  e.  _V )
33 forn 5220 . . . . . 6  |-  ( F : A -onto-> B  ->  ran  F  =  B )
3433eleq1d 2156 . . . . 5  |-  ( F : A -onto-> B  -> 
( ran  F  e.  _V 
<->  B  e.  _V )
)
3532, 34syl5ibcom 153 . . . 4  |-  ( F  e.  _V  ->  ( F : A -onto-> B  ->  B  e.  _V )
)
3635imp 122 . . 3  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  B  e.  _V )
37 pwexg 4007 . . 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 5184 . . . 4  |-  ( ( F  e.  _V  /\  F : A --> B )  ->  A  e.  _V )
403, 39sylan2 280 . . 3  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  A  e.  _V )
41 pwexg 4007 . . 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 6471 1  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ~P B  ~<_  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   _Vcvv 2619    C_ wss 2997   ~Pcpw 3425   class class class wbr 3837   `'ccnv 4427   dom cdm 4428   ran crn 4429   "cima 4431   -->wf 4998   -onto->wfo 5000    ~<_ cdom 6436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-fv 5010  df-dom 6439
This theorem is referenced by: (None)
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