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Theorem wdomima2g 7487
Description: A set is weakly dominant over its image under any function. This version of wdomimag 7488 is stated so as to avoid ax-rep 4261. (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 4831 . 2  |-  ( F
" A )  =  ran  ( F  |`  A )
2 funres 5432 . . . . . . . 8  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
3 funforn 5600 . . . . . . . 8  |-  ( Fun  ( F  |`  A )  <-> 
( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )
42, 3sylib 189 . . . . . . 7  |-  ( Fun 
F  ->  ( F  |`  A ) : dom  ( F  |`  A )
-onto->
ran  ( F  |`  A ) )
543ad2ant1 978 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )
6 fof 5593 . . . . . 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 5107 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
9 inss1 3504 . . . . . . 7  |-  ( A  i^i  dom  F )  C_  A
108, 9eqsstri 3321 . . . . . 6  |-  dom  ( F  |`  A )  C_  A
11 simp2 958 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  A  e.  V )
12 ssexg 4290 . . . . . 6  |-  ( ( dom  ( F  |`  A )  C_  A  /\  A  e.  V
)  ->  dom  ( F  |`  A )  e.  _V )
1310, 11, 12sylancr 645 . . . . 5  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  dom  ( F  |`  A )  e.  _V )
14 simp3 959 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F " A )  e.  W )
151, 14syl5eqelr 2472 . . . . 5  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ran  ( F  |`  A )  e.  W )
16 fex2 5543 . . . . 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 1184 . . . 4  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F  |`  A )  e. 
_V )
18 fowdom 7472 . . . 4  |-  ( ( ( F  |`  A )  e.  _V  /\  ( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )  ->  ran  ( F  |`  A )  ~<_*  dom  ( F  |`  A ) )
1917, 5, 18syl2anc 643 . . 3  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ran  ( F  |`  A )  ~<_*  dom  ( F  |`  A ) )
20 ssdomg 7089 . . . . . 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 7475 . . . . 5  |-  ( dom  ( F  |`  A )  ~<_  A  ->  dom  ( F  |`  A )  ~<_*  A )
2321, 22syl 16 . . . 4  |-  ( A  e.  V  ->  dom  ( F  |`  A )  ~<_*  A )
24233ad2ant2 979 . . 3  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  dom  ( F  |`  A )  ~<_*  A )
25 wdomtr 7476 . . 3  |-  ( ( ran  ( F  |`  A )  ~<_*  dom  ( F  |`  A )  /\  dom  ( F  |`  A )  ~<_*  A )  ->  ran  ( F  |`  A )  ~<_*  A )
2619, 24, 25syl2anc 643 . 2  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ran  ( F  |`  A )  ~<_*  A )
271, 26syl5eqbr 4186 1  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F " A )  ~<_*  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    e. wcel 1717   _Vcvv 2899    i^i cin 3262    C_ wss 3263   class class class wbr 4153   dom cdm 4818   ran crn 4819    |` cres 4820   "cima 4821   Fun wfun 5388   -->wf 5390   -onto->wfo 5392    ~<_ cdom 7043    ~<_* cwdom 7458
This theorem is referenced by:  wdomimag  7488  unxpwdom2  7489
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-wdom 7460
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