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Theorem wdomima2g 7546
Description: A set is weakly dominant over its image under any function. This version of wdomimag 7547 is stated so as to avoid ax-rep 4312. (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 4883 . 2  |-  ( F
" A )  =  ran  ( F  |`  A )
2 funres 5484 . . . . . . . 8  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
3 funforn 5652 . . . . . . . 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 5645 . . . . . 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 5159 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
9 inss1 3553 . . . . . . 7  |-  ( A  i^i  dom  F )  C_  A
108, 9eqsstri 3370 . . . . . 6  |-  dom  ( F  |`  A )  C_  A
11 simp2 958 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  A  e.  V )
12 ssexg 4341 . . . . . 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 2520 . . . . 5  |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ran  ( F  |`  A )  e.  W )
16 fex2 5595 . . . . 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 7531 . . . 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 7145 . . . . . 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 7534 . . . . 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 7535 . . 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 4237 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 1725   _Vcvv 2948    i^i cin 3311    C_ wss 3312   class class class wbr 4204   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Fun wfun 5440   -->wf 5442   -onto->wfo 5444    ~<_ cdom 7099    ~<_* cwdom 7517
This theorem is referenced by:  wdomimag  7547  unxpwdom2  7548
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-wdom 7519
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