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Theorem imadomg 8043
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 4601 . . . 4  |-  ( F
" A )  =  ran  (  F  |`  A )
2 resfunexg 5589 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F  |`  A )  e. 
_V )
3 dmexg 4846 . . . . . 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 5150 . . . . . . 7  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
6 funforn 5315 . . . . . . 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 8034 . . . . 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 3953 . . 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 4883 . . . . . 6  |-  dom  (  F  |`  A )  =  ( A  i^i  dom  F )
14 inss1 3296 . . . . . 6  |-  ( A  i^i  dom  F )  C_  A
1513, 14eqsstri 3129 . . . . 5  |-  dom  (  F  |`  A )  C_  A
16 ssdomg 6793 . . . . 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 6799 . . . 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 2727    i^i cin 3077    C_ wss 3078   class class class wbr 3920   dom cdm 4580   ran crn 4581    |` cres 4582   "cima 4583   Fun wfun 4586   -onto->wfo 4590    ~<_ cdom 6747
This theorem is referenced by:  uniimadom  8050  hausmapdom  17058
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-ac2 7973
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-card 7456  df-acn 7459  df-ac 7627
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