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Theorem imadomg 8127
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 4682 . . . 4  |-  ( F
" A )  =  ran  (  F  |`  A )
2 resfunexg 5671 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F  |`  A )  e. 
_V )
3 dmexg 4927 . . . . . 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 5231 . . . . . . 7  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
6 funforn 5396 . . . . . . 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 8118 . . . . 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 4030 . . 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 4964 . . . . . 6  |-  dom  (  F  |`  A )  =  ( A  i^i  dom  F )
14 inss1 3364 . . . . . 6  |-  ( A  i^i  dom  F )  C_  A
1513, 14eqsstri 3183 . . . . 5  |-  dom  (  F  |`  A )  C_  A
16 ssdomg 6875 . . . . 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 6882 . . . 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 1621   _Vcvv 2763    i^i cin 3126    C_ wss 3127   class class class wbr 3997   dom cdm 4661   ran crn 4662    |` cres 4663   "cima 4664   Fun wfun 4667   -onto->wfo 4671    ~<_ cdom 6829
This theorem is referenced by:  uniimadom  8134  hausmapdom  17188
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-ac2 8057
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 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-suc 4370  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-card 7540  df-acn 7543  df-ac 7711
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