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Theorem wdomima2g 8110
Description: A set is weakly dominant over its image under any function. This version of wdomimag 8111 is stated so as to avoid ax-rep 4536. (Contributed by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
wdomima2g  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F " A )  ~<_*  A
)

Proof of Theorem wdomima2g
StepHypRef Expression
1 df-ima 4866 . 2  |-  ( F
" A )  =  ran  ( F  |`  A )
2 funres 5640 . . . . . . . 8  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
3 funforn 5817 . . . . . . . 8  |-  ( Fun  ( F  |`  A )  <-> 
( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )
42, 3sylib 199 . . . . . . 7  |-  ( Fun 
F  ->  ( F  |`  A ) : dom  ( F  |`  A )
-onto->
ran  ( F  |`  A ) )
543ad2ant1 1026 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )
6 fof 5810 . . . . . 6  |-  ( ( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A )  -> 
( F  |`  A ) : dom  ( F  |`  A ) --> ran  ( F  |`  A ) )
75, 6syl 17 . . . . 5  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F  |`  A ) : dom  ( F  |`  A ) --> ran  ( F  |`  A ) )
8 dmres 5144 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
9 inss1 3682 . . . . . . 7  |-  ( A  i^i  dom  F )  C_  A
108, 9eqsstri 3494 . . . . . 6  |-  dom  ( F  |`  A )  C_  A
11 simp2 1006 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  A  e.  V )
12 ssexg 4570 . . . . . 6  |-  ( ( dom  ( F  |`  A )  C_  A  /\  A  e.  V
)  ->  dom  ( F  |`  A )  e.  _V )
1310, 11, 12sylancr 667 . . . . 5  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  dom  ( F  |`  A )  e.  _V )
14 simp3 1007 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F " A )  e.  W )
151, 14syl5eqelr 2512 . . . . 5  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ran  ( F  |`  A )  e.  W )
16 fex2 6762 . . . . 5  |-  ( ( ( F  |`  A ) : dom  ( F  |`  A ) --> ran  ( F  |`  A )  /\  dom  ( F  |`  A )  e.  _V  /\  ran  ( F  |`  A )  e.  W )  -> 
( F  |`  A )  e.  _V )
177, 13, 15, 16syl3anc 1264 . . . 4  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F  |`  A )  e. 
_V )
18 fowdom 8095 . . . 4  |-  ( ( ( F  |`  A )  e.  _V  /\  ( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )  ->  ran  ( F  |`  A )  ~<_*  dom  ( F  |`  A ) )
1917, 5, 18syl2anc 665 . . 3  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ran  ( F  |`  A )  ~<_*  dom  ( F  |`  A ) )
20 ssdomg 7625 . . . . . 6  |-  ( A  e.  V  ->  ( dom  ( F  |`  A ) 
C_  A  ->  dom  ( F  |`  A )  ~<_  A ) )
2110, 20mpi 20 . . . . 5  |-  ( A  e.  V  ->  dom  ( F  |`  A )  ~<_  A )
22 domwdom 8098 . . . . 5  |-  ( dom  ( F  |`  A )  ~<_  A  ->  dom  ( F  |`  A )  ~<_*  A )
2321, 22syl 17 . . . 4  |-  ( A  e.  V  ->  dom  ( F  |`  A )  ~<_*  A )
24233ad2ant2 1027 . . 3  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  dom  ( F  |`  A )  ~<_*  A )
25 wdomtr 8099 . . 3  |-  ( ( ran  ( F  |`  A )  ~<_*  dom  ( F  |`  A )  /\  dom  ( F  |`  A )  ~<_*  A )  ->  ran  ( F  |`  A )  ~<_*  A )
2619, 24, 25syl2anc 665 . 2  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ran  ( F  |`  A )  ~<_*  A )
271, 26syl5eqbr 4457 1  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F " A )  ~<_*  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    e. wcel 1872   _Vcvv 3080    i^i cin 3435    C_ wss 3436   class class class wbr 4423   dom cdm 4853   ran crn 4854    |` cres 4855   "cima 4856   Fun wfun 5595   -->wf 5597   -onto->wfo 5599    ~<_ cdom 7578    ~<_* cwdom 8081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-wdom 8083
This theorem is referenced by:  wdomimag  8111  unxpwdom2  8112
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