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Theorem fopwdom 6986
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 5041 . . . . . 6  |-  ( `' F " a ) 
C_  ran  `' F
2 dfdm4 4888 . . . . . . 7  |-  dom  F  =  ran  `' F
3 fof 5467 . . . . . . . 8  |-  ( F : A -onto-> B  ->  F : A --> B )
4 fdm 5409 . . . . . . . 8  |-  ( F : A --> B  ->  dom  F  =  A )
53, 4syl 15 . . . . . . 7  |-  ( F : A -onto-> B  ->  dom  F  =  A )
62, 5syl5eqr 2342 . . . . . 6  |-  ( F : A -onto-> B  ->  ran  `' F  =  A
)
71, 6syl5sseq 3239 . . . . 5  |-  ( F : A -onto-> B  -> 
( `' F "
a )  C_  A
)
87adantl 452 . . . 4  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( `' F " a )  C_  A
)
9 cnvexg 5224 . . . . . 6  |-  ( F  e.  _V  ->  `' F  e.  _V )
109adantr 451 . . . . 5  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  `' F  e. 
_V )
11 imaexg 5042 . . . . 5  |-  ( `' F  e.  _V  ->  ( `' F " a )  e.  _V )
12 elpwg 3645 . . . . 5  |-  ( ( `' F " a )  e.  _V  ->  (
( `' F "
a )  e.  ~P A 
<->  ( `' F "
a )  C_  A
) )
1310, 11, 123syl 18 . . . 4  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( ( `' F " a )  e.  ~P A  <->  ( `' F " a )  C_  A ) )
148, 13mpbird 223 . . 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 5024 . . . . . . 7  |-  ( ( `' F " a )  =  ( `' F " b )  ->  ( F " ( `' F " a ) )  =  ( F " ( `' F " b ) ) )
1716adantl 452 . . . . . 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 735 . . . . . . 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 736 . . . . . . . 8  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
a  e.  ~P B
)
20 vex 2804 . . . . . . . . 9  |-  a  e. 
_V
2120elpw 3644 . . . . . . . 8  |-  ( a  e.  ~P B  <->  a  C_  B )
2219, 21sylib 188 . . . . . . 7  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
a  C_  B )
23 foimacnv 5506 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  a  C_  B )  ->  ( F "
( `' F "
a ) )  =  a )
2418, 22, 23syl2anc 642 . . . . . 6  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
( F " ( `' F " a ) )  =  a )
25 simplrr 737 . . . . . . . 8  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
b  e.  ~P B
)
26 vex 2804 . . . . . . . . 9  |-  b  e. 
_V
2726elpw 3644 . . . . . . . 8  |-  ( b  e.  ~P B  <->  b  C_  B )
2825, 27sylib 188 . . . . . . 7  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
b  C_  B )
29 foimacnv 5506 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  b  C_  B )  ->  ( F "
( `' F "
b ) )  =  b )
3018, 28, 29syl2anc 642 . . . . . 6  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
( F " ( `' F " b ) )  =  b )
3117, 24, 303eqtr3d 2336 . . . . 5  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
a  =  b )
3231ex 423 . . . 4  |-  ( ( ( F  e.  _V  /\  F : A -onto-> B
)  /\  ( a  e.  ~P B  /\  b  e.  ~P B ) )  ->  ( ( `' F " a )  =  ( `' F " b )  ->  a  =  b ) )
33 imaeq2 5024 . . . 4  |-  ( a  =  b  ->  ( `' F " a )  =  ( `' F " b ) )
3432, 33impbid1 194 . . 3  |-  ( ( ( F  e.  _V  /\  F : A -onto-> B
)  /\  ( a  e.  ~P B  /\  b  e.  ~P B ) )  ->  ( ( `' F " a )  =  ( `' F " b )  <->  a  =  b ) )
3534ex 423 . 2  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( ( a  e.  ~P B  /\  b  e.  ~P B
)  ->  ( ( `' F " a )  =  ( `' F " b )  <->  a  =  b ) ) )
36 rnexg 4956 . . . . 5  |-  ( F  e.  _V  ->  ran  F  e.  _V )
37 forn 5470 . . . . . 6  |-  ( F : A -onto-> B  ->  ran  F  =  B )
3837eleq1d 2362 . . . . 5  |-  ( F : A -onto-> B  -> 
( ran  F  e.  _V 
<->  B  e.  _V )
)
3936, 38syl5ibcom 211 . . . 4  |-  ( F  e.  _V  ->  ( F : A -onto-> B  ->  B  e.  _V )
)
4039imp 418 . . 3  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  B  e.  _V )
41 pwexg 4210 . . 3  |-  ( B  e.  _V  ->  ~P B  e.  _V )
4240, 41syl 15 . 2  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ~P B  e. 
_V )
43 dmfex 5440 . . . 4  |-  ( ( F  e.  _V  /\  F : A --> B )  ->  A  e.  _V )
443, 43sylan2 460 . . 3  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  A  e.  _V )
45 pwexg 4210 . . 3  |-  ( A  e.  _V  ->  ~P A  e.  _V )
4644, 45syl 15 . 2  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ~P A  e. 
_V )
4715, 35, 42, 46dom3d 6919 1  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ~P B  ~<_  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   class class class wbr 4039   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708   -->wf 5267   -onto->wfo 5269    ~<_ cdom 6877
This theorem is referenced by:  pwdom  7029  wdompwdom  7308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-fv 5279  df-dom 6881
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