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Theorem wdomima2g 8012
Description: A set is weakly dominant over its image under any function. This version of wdomimag 8013 is stated so as to avoid ax-rep 4558. (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 5012 . 2  |-  ( F
" A )  =  ran  ( F  |`  A )
2 funres 5627 . . . . . . . 8  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
3 funforn 5802 . . . . . . . 8  |-  ( Fun  ( F  |`  A )  <-> 
( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )
42, 3sylib 196 . . . . . . 7  |-  ( Fun 
F  ->  ( F  |`  A ) : dom  ( F  |`  A )
-onto->
ran  ( F  |`  A ) )
543ad2ant1 1017 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )
6 fof 5795 . . . . . 6  |-  ( ( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A )  -> 
( F  |`  A ) : dom  ( F  |`  A ) --> ran  ( F  |`  A ) )
75, 6syl 16 . . . . 5  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F  |`  A ) : dom  ( F  |`  A ) --> ran  ( F  |`  A ) )
8 dmres 5294 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
9 inss1 3718 . . . . . . 7  |-  ( A  i^i  dom  F )  C_  A
108, 9eqsstri 3534 . . . . . 6  |-  dom  ( F  |`  A )  C_  A
11 simp2 997 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  A  e.  V )
12 ssexg 4593 . . . . . 6  |-  ( ( dom  ( F  |`  A )  C_  A  /\  A  e.  V
)  ->  dom  ( F  |`  A )  e.  _V )
1310, 11, 12sylancr 663 . . . . 5  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  dom  ( F  |`  A )  e.  _V )
14 simp3 998 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F " A )  e.  W )
151, 14syl5eqelr 2560 . . . . 5  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ran  ( F  |`  A )  e.  W )
16 fex2 6739 . . . . 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 1228 . . . 4  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F  |`  A )  e. 
_V )
18 fowdom 7997 . . . 4  |-  ( ( ( F  |`  A )  e.  _V  /\  ( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )  ->  ran  ( F  |`  A )  ~<_*  dom  ( F  |`  A ) )
1917, 5, 18syl2anc 661 . . 3  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ran  ( F  |`  A )  ~<_*  dom  ( F  |`  A ) )
20 ssdomg 7561 . . . . . 6  |-  ( A  e.  V  ->  ( dom  ( F  |`  A ) 
C_  A  ->  dom  ( F  |`  A )  ~<_  A ) )
2110, 20mpi 17 . . . . 5  |-  ( A  e.  V  ->  dom  ( F  |`  A )  ~<_  A )
22 domwdom 8000 . . . . 5  |-  ( dom  ( F  |`  A )  ~<_  A  ->  dom  ( F  |`  A )  ~<_*  A )
2321, 22syl 16 . . . 4  |-  ( A  e.  V  ->  dom  ( F  |`  A )  ~<_*  A )
24233ad2ant2 1018 . . 3  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  dom  ( F  |`  A )  ~<_*  A )
25 wdomtr 8001 . . 3  |-  ( ( ran  ( F  |`  A )  ~<_*  dom  ( F  |`  A )  /\  dom  ( F  |`  A )  ~<_*  A )  ->  ran  ( F  |`  A )  ~<_*  A )
2619, 24, 25syl2anc 661 . 2  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ran  ( F  |`  A )  ~<_*  A )
271, 26syl5eqbr 4480 1  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F " A )  ~<_*  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    e. wcel 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476   class class class wbr 4447   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002   Fun wfun 5582   -->wf 5584   -onto->wfo 5586    ~<_ cdom 7514    ~<_* cwdom 7983
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-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-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-wdom 7985
This theorem is referenced by:  wdomimag  8013  unxpwdom2  8014
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