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Theorem wdomima2g 7302
Description: A set is weakly dominant over its image under any function. This version of wdomimag 7303 is stated so as to avoid ax-rep 4133. (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 4704 . 2  |-  ( F
" A )  =  ran  ( F  |`  A )
2 funres 5295 . . . . . . . 8  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
3 funforn 5460 . . . . . . . 8  |-  ( Fun  ( F  |`  A )  <-> 
( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )
42, 3sylib 188 . . . . . . 7  |-  ( Fun 
F  ->  ( F  |`  A ) : dom  ( F  |`  A )
-onto->
ran  ( F  |`  A ) )
543ad2ant1 976 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )
6 fof 5453 . . . . . 6  |-  ( ( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A )  -> 
( F  |`  A ) : dom  ( F  |`  A ) --> ran  ( F  |`  A ) )
75, 6syl 15 . . . . 5  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F  |`  A ) : dom  ( F  |`  A ) --> ran  ( F  |`  A ) )
8 dmres 4978 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
9 inss1 3391 . . . . . . 7  |-  ( A  i^i  dom  F )  C_  A
108, 9eqsstri 3210 . . . . . 6  |-  dom  ( F  |`  A )  C_  A
11 simp2 956 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  A  e.  V )
12 ssexg 4162 . . . . . 6  |-  ( ( dom  ( F  |`  A )  C_  A  /\  A  e.  V
)  ->  dom  ( F  |`  A )  e.  _V )
1310, 11, 12sylancr 644 . . . . 5  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  dom  ( F  |`  A )  e.  _V )
14 simp3 957 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F " A )  e.  W )
151, 14syl5eqelr 2370 . . . . 5  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ran  ( F  |`  A )  e.  W )
16 fex2 5403 . . . . 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 1182 . . . 4  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F  |`  A )  e. 
_V )
18 fowdom 7287 . . . 4  |-  ( ( ( F  |`  A )  e.  _V  /\  ( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )  ->  ran  ( F  |`  A )  ~<_*  dom  ( F  |`  A ) )
1917, 5, 18syl2anc 642 . . 3  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ran  ( F  |`  A )  ~<_*  dom  ( F  |`  A ) )
20 ssdomg 6909 . . . . . 6  |-  ( A  e.  V  ->  ( dom  ( F  |`  A ) 
C_  A  ->  dom  ( F  |`  A )  ~<_  A ) )
2110, 20mpi 16 . . . . 5  |-  ( A  e.  V  ->  dom  ( F  |`  A )  ~<_  A )
22 domwdom 7290 . . . . 5  |-  ( dom  ( F  |`  A )  ~<_  A  ->  dom  ( F  |`  A )  ~<_*  A )
2321, 22syl 15 . . . 4  |-  ( A  e.  V  ->  dom  ( F  |`  A )  ~<_*  A )
24233ad2ant2 977 . . 3  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  dom  ( F  |`  A )  ~<_*  A )
25 wdomtr 7291 . . 3  |-  ( ( ran  ( F  |`  A )  ~<_*  dom  ( F  |`  A )  /\  dom  ( F  |`  A )  ~<_*  A )  ->  ran  ( F  |`  A )  ~<_*  A )
2619, 24, 25syl2anc 642 . 2  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ran  ( F  |`  A )  ~<_*  A )
271, 26syl5eqbr 4058 1  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F " A )  ~<_*  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1686   _Vcvv 2790    i^i cin 3153    C_ wss 3154   class class class wbr 4025   dom cdm 4691   ran crn 4692    |` cres 4693   "cima 4694   Fun wfun 5251   -->wf 5253   -onto->wfo 5255    ~<_ cdom 6863    ~<_* cwdom 7273
This theorem is referenced by:  wdomimag  7303  unxpwdom2  7304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-wdom 7275
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