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Theorem fopwdom 7216
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 5216 . . . . . 6  |-  ( `' F " a ) 
C_  ran  `' F
2 dfdm4 5063 . . . . . . 7  |-  dom  F  =  ran  `' F
3 fof 5653 . . . . . . . 8  |-  ( F : A -onto-> B  ->  F : A --> B )
4 fdm 5595 . . . . . . . 8  |-  ( F : A --> B  ->  dom  F  =  A )
53, 4syl 16 . . . . . . 7  |-  ( F : A -onto-> B  ->  dom  F  =  A )
62, 5syl5eqr 2482 . . . . . 6  |-  ( F : A -onto-> B  ->  ran  `' F  =  A
)
71, 6syl5sseq 3396 . . . . 5  |-  ( F : A -onto-> B  -> 
( `' F "
a )  C_  A
)
87adantl 453 . . . 4  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( `' F " a )  C_  A
)
9 cnvexg 5405 . . . . . 6  |-  ( F  e.  _V  ->  `' F  e.  _V )
109adantr 452 . . . . 5  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  `' F  e. 
_V )
11 imaexg 5217 . . . . 5  |-  ( `' F  e.  _V  ->  ( `' F " a )  e.  _V )
12 elpwg 3806 . . . . 5  |-  ( ( `' F " a )  e.  _V  ->  (
( `' F "
a )  e.  ~P A 
<->  ( `' F "
a )  C_  A
) )
1310, 11, 123syl 19 . . . 4  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( ( `' F " a )  e.  ~P A  <->  ( `' F " a )  C_  A ) )
148, 13mpbird 224 . . 3  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( `' F " a )  e.  ~P A )
1514a1d 23 . 2  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( a  e. 
~P B  ->  ( `' F " a )  e.  ~P A ) )
16 imaeq2 5199 . . . . . . 7  |-  ( ( `' F " a )  =  ( `' F " b )  ->  ( F " ( `' F " a ) )  =  ( F " ( `' F " b ) ) )
1716adantl 453 . . . . . 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 736 . . . . . . 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 737 . . . . . . . 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 3808 . . . . . . 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 5692 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  a  C_  B )  ->  ( F "
( `' F "
a ) )  =  a )
2218, 20, 21syl2anc 643 . . . . . 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 738 . . . . . . . 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 3808 . . . . . . 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 5692 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  b  C_  B )  ->  ( F "
( `' F "
b ) )  =  b )
2618, 24, 25syl2anc 643 . . . . . 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 2476 . . . . 5  |-  ( ( ( ( F  e. 
_V  /\  F : A -onto-> B )  /\  (
a  e.  ~P B  /\  b  e.  ~P B ) )  /\  ( `' F " a )  =  ( `' F " b ) )  -> 
a  =  b )
2827ex 424 . . . 4  |-  ( ( ( F  e.  _V  /\  F : A -onto-> B
)  /\  ( a  e.  ~P B  /\  b  e.  ~P B ) )  ->  ( ( `' F " a )  =  ( `' F " b )  ->  a  =  b ) )
29 imaeq2 5199 . . . 4  |-  ( a  =  b  ->  ( `' F " a )  =  ( `' F " b ) )
3028, 29impbid1 195 . . 3  |-  ( ( ( F  e.  _V  /\  F : A -onto-> B
)  /\  ( a  e.  ~P B  /\  b  e.  ~P B ) )  ->  ( ( `' F " a )  =  ( `' F " b )  <->  a  =  b ) )
3130ex 424 . 2  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ( ( a  e.  ~P B  /\  b  e.  ~P B
)  ->  ( ( `' F " a )  =  ( `' F " b )  <->  a  =  b ) ) )
32 rnexg 5131 . . . . 5  |-  ( F  e.  _V  ->  ran  F  e.  _V )
33 forn 5656 . . . . . 6  |-  ( F : A -onto-> B  ->  ran  F  =  B )
3433eleq1d 2502 . . . . 5  |-  ( F : A -onto-> B  -> 
( ran  F  e.  _V 
<->  B  e.  _V )
)
3532, 34syl5ibcom 212 . . . 4  |-  ( F  e.  _V  ->  ( F : A -onto-> B  ->  B  e.  _V )
)
3635imp 419 . . 3  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  B  e.  _V )
37 pwexg 4383 . . 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 5626 . . . 4  |-  ( ( F  e.  _V  /\  F : A --> B )  ->  A  e.  _V )
403, 39sylan2 461 . . 3  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  A  e.  _V )
41 pwexg 4383 . . 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 7149 1  |-  ( ( F  e.  _V  /\  F : A -onto-> B )  ->  ~P B  ~<_  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    C_ wss 3320   ~Pcpw 3799   class class class wbr 4212   `'ccnv 4877   dom cdm 4878   ran crn 4879   "cima 4881   -->wf 5450   -onto->wfo 5452    ~<_ cdom 7107
This theorem is referenced by:  pwdom  7259  wdompwdom  7546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-fv 5462  df-dom 7111
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