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Theorem imadomg 8154
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 4701 . . . 4  |-  ( F
" A )  =  ran  (  F  |`  A )
2 resfunexg 5698 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F  |`  A )  e. 
_V )
3 dmexg 4938 . . . . . 6  |-  ( ( F  |`  A )  e.  _V  ->  dom  (  F  |`  A )  e.  _V )
42, 3syl 17 . . . . 5  |-  ( ( Fun  F  /\  A  e.  B )  ->  dom  (  F  |`  A )  e.  _V )
5 funres 5258 . . . . . . 7  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
6 funforn 5423 . . . . . . 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 8145 . . . . 5  |-  ( dom  (  F  |`  A )  e.  _V  ->  (
( F  |`  A ) : dom  (  F  |`  A ) -onto-> ran  (  F  |`  A )  ->  ran  (  F  |`  A )  ~<_  dom  (  F  |`  A ) ) )
104, 8, 9sylc 58 . . . 4  |-  ( ( Fun  F  /\  A  e.  B )  ->  ran  (  F  |`  A )  ~<_  dom  (  F  |`  A ) )
111, 10syl5eqbr 4057 . . 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 4975 . . . . . 6  |-  dom  (  F  |`  A )  =  ( A  i^i  dom  F )
14 inss1 3390 . . . . . 6  |-  ( A  i^i  dom  F )  C_  A
1513, 14eqsstri 3209 . . . . 5  |-  dom  (  F  |`  A )  C_  A
16 ssdomg 6902 . . . . 5  |-  ( A  e.  B  ->  ( dom  (  F  |`  A ) 
C_  A  ->  dom  (  F  |`  A )  ~<_  A ) )
1715, 16mpi 18 . . . 4  |-  ( A  e.  B  ->  dom  (  F  |`  A )  ~<_  A )
18 domtr 6909 . . . 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 42 1  |-  ( A  e.  B  ->  ( Fun  F  ->  ( F " A )  ~<_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1688   _Vcvv 2789    i^i cin 3152    C_ wss 3153   class class class wbr 4024   dom cdm 4688   ran crn 4689    |` cres 4690   "cima 4691   Fun wfun 5215   -onto->wfo 5219    ~<_ cdom 6856
This theorem is referenced by:  uniimadom  8161  hausmapdom  17220
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-ac2 8084
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-card 7567  df-acn 7570  df-ac 7738
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