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Theorem imadomg 8413
Description: An image of a function under a set is dominated by the set. Proposition 10.34 of [TakeutiZaring] p. 92. (Contributed by NM, 23-Jul-2004.)
Assertion
Ref Expression
imadomg  |-  ( A  e.  B  ->  ( Fun  F  ->  ( F " A )  ~<_  A ) )

Proof of Theorem imadomg
StepHypRef Expression
1 df-ima 4892 . . . 4  |-  ( F
" A )  =  ran  ( F  |`  A )
2 resfunexg 5958 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F  |`  A )  e. 
_V )
3 dmexg 5131 . . . . . 6  |-  ( ( F  |`  A )  e.  _V  ->  dom  ( F  |`  A )  e.  _V )
42, 3syl 16 . . . . 5  |-  ( ( Fun  F  /\  A  e.  B )  ->  dom  ( F  |`  A )  e.  _V )
5 funres 5493 . . . . . . 7  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
6 funforn 5661 . . . . . . 7  |-  ( Fun  ( F  |`  A )  <-> 
( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )
75, 6sylib 190 . . . . . 6  |-  ( Fun 
F  ->  ( F  |`  A ) : dom  ( F  |`  A )
-onto->
ran  ( F  |`  A ) )
87adantr 453 . . . . 5  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )
9 fodomg 8404 . . . . 5  |-  ( dom  ( F  |`  A )  e.  _V  ->  (
( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A )  ->  ran  ( F  |`  A )  ~<_  dom  ( F  |`  A ) ) )
104, 8, 9sylc 59 . . . 4  |-  ( ( Fun  F  /\  A  e.  B )  ->  ran  ( F  |`  A )  ~<_  dom  ( F  |`  A ) )
111, 10syl5eqbr 4246 . . 3  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F " A )  ~<_  dom  ( F  |`  A ) )
1211expcom 426 . 2  |-  ( A  e.  B  ->  ( Fun  F  ->  ( F " A )  ~<_  dom  ( F  |`  A ) ) )
13 dmres 5168 . . . . . 6  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
14 inss1 3562 . . . . . 6  |-  ( A  i^i  dom  F )  C_  A
1513, 14eqsstri 3379 . . . . 5  |-  dom  ( F  |`  A )  C_  A
16 ssdomg 7154 . . . . 5  |-  ( A  e.  B  ->  ( dom  ( F  |`  A ) 
C_  A  ->  dom  ( F  |`  A )  ~<_  A ) )
1715, 16mpi 17 . . . 4  |-  ( A  e.  B  ->  dom  ( F  |`  A )  ~<_  A )
18 domtr 7161 . . . 4  |-  ( ( ( F " A
)  ~<_  dom  ( F  |`  A )  /\  dom  ( F  |`  A )  ~<_  A )  ->  ( F " A )  ~<_  A )
1917, 18sylan2 462 . . 3  |-  ( ( ( F " A
)  ~<_  dom  ( F  |`  A )  /\  A  e.  B )  ->  ( F " A )  ~<_  A )
2019expcom 426 . 2  |-  ( A  e.  B  ->  (
( F " A
)  ~<_  dom  ( F  |`  A )  ->  ( F " A )  ~<_  A ) )
2112, 20syld 43 1  |-  ( A  e.  B  ->  ( Fun  F  ->  ( F " A )  ~<_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   _Vcvv 2957    i^i cin 3320    C_ wss 3321   class class class wbr 4213   dom cdm 4879   ran crn 4880    |` cres 4881   "cima 4882   Fun wfun 5449   -onto->wfo 5453    ~<_ cdom 7108
This theorem is referenced by:  uniimadom  8420  hausmapdom  17564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-ac2 8344
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-suc 4588  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-card 7827  df-acn 7830  df-ac 7998
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