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Theorem imadomg 8159
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 4702 . . . 4  |-  ( F
" A )  =  ran  ( F  |`  A )
2 resfunexg 5737 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F  |`  A )  e. 
_V )
3 dmexg 4939 . . . . . 6  |-  ( ( F  |`  A )  e.  _V  ->  dom  ( F  |`  A )  e.  _V )
42, 3syl 15 . . . . 5  |-  ( ( Fun  F  /\  A  e.  B )  ->  dom  ( F  |`  A )  e.  _V )
5 funres 5293 . . . . . . 7  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
6 funforn 5458 . . . . . . 7  |-  ( Fun  ( F  |`  A )  <-> 
( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )
75, 6sylib 188 . . . . . 6  |-  ( Fun 
F  ->  ( F  |`  A ) : dom  ( F  |`  A )
-onto->
ran  ( F  |`  A ) )
87adantr 451 . . . . 5  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )
9 fodomg 8150 . . . . 5  |-  ( dom  ( F  |`  A )  e.  _V  ->  (
( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A )  ->  ran  ( F  |`  A )  ~<_  dom  ( F  |`  A ) ) )
104, 8, 9sylc 56 . . . 4  |-  ( ( Fun  F  /\  A  e.  B )  ->  ran  ( F  |`  A )  ~<_  dom  ( F  |`  A ) )
111, 10syl5eqbr 4056 . . 3  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F " A )  ~<_  dom  ( F  |`  A ) )
1211expcom 424 . 2  |-  ( A  e.  B  ->  ( Fun  F  ->  ( F " A )  ~<_  dom  ( F  |`  A ) ) )
13 dmres 4976 . . . . . 6  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
14 inss1 3389 . . . . . 6  |-  ( A  i^i  dom  F )  C_  A
1513, 14eqsstri 3208 . . . . 5  |-  dom  ( F  |`  A )  C_  A
16 ssdomg 6907 . . . . 5  |-  ( A  e.  B  ->  ( dom  ( F  |`  A ) 
C_  A  ->  dom  ( F  |`  A )  ~<_  A ) )
1715, 16mpi 16 . . . 4  |-  ( A  e.  B  ->  dom  ( F  |`  A )  ~<_  A )
18 domtr 6914 . . . 4  |-  ( ( ( F " A
)  ~<_  dom  ( F  |`  A )  /\  dom  ( F  |`  A )  ~<_  A )  ->  ( F " A )  ~<_  A )
1917, 18sylan2 460 . . 3  |-  ( ( ( F " A
)  ~<_  dom  ( F  |`  A )  /\  A  e.  B )  ->  ( F " A )  ~<_  A )
2019expcom 424 . 2  |-  ( A  e.  B  ->  (
( F " A
)  ~<_  dom  ( F  |`  A )  ->  ( F " A )  ~<_  A ) )
2112, 20syld 40 1  |-  ( A  e.  B  ->  ( Fun  F  ->  ( F " A )  ~<_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   class class class wbr 4023   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249   -onto->wfo 5253    ~<_ cdom 6861
This theorem is referenced by:  uniimadom  8166  hausmapdom  17226
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-ac2 8089
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-card 7572  df-acn 7575  df-ac 7743
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