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Theorem imadomg 8722
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 4874 . . . 4  |-  ( F
" A )  =  ran  ( F  |`  A )
2 resfunexg 5964 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F  |`  A )  e. 
_V )
3 dmexg 6530 . . . . . 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 5478 . . . . . . 7  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
6 funforn 5648 . . . . . . 7  |-  ( Fun  ( F  |`  A )  <-> 
( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )
75, 6sylib 196 . . . . . 6  |-  ( Fun 
F  ->  ( F  |`  A ) : dom  ( F  |`  A )
-onto->
ran  ( F  |`  A ) )
87adantr 465 . . . . 5  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )
9 fodomg 8713 . . . . 5  |-  ( dom  ( F  |`  A )  e.  _V  ->  (
( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A )  ->  ran  ( F  |`  A )  ~<_  dom  ( F  |`  A ) ) )
104, 8, 9sylc 60 . . . 4  |-  ( ( Fun  F  /\  A  e.  B )  ->  ran  ( F  |`  A )  ~<_  dom  ( F  |`  A ) )
111, 10syl5eqbr 4346 . . 3  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F " A )  ~<_  dom  ( F  |`  A ) )
1211expcom 435 . 2  |-  ( A  e.  B  ->  ( Fun  F  ->  ( F " A )  ~<_  dom  ( F  |`  A ) ) )
13 dmres 5152 . . . . . 6  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
14 inss1 3591 . . . . . 6  |-  ( A  i^i  dom  F )  C_  A
1513, 14eqsstri 3407 . . . . 5  |-  dom  ( F  |`  A )  C_  A
16 ssdomg 7376 . . . . 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 7383 . . . 4  |-  ( ( ( F " A
)  ~<_  dom  ( F  |`  A )  /\  dom  ( F  |`  A )  ~<_  A )  ->  ( F " A )  ~<_  A )
1917, 18sylan2 474 . . 3  |-  ( ( ( F " A
)  ~<_  dom  ( F  |`  A )  /\  A  e.  B )  ->  ( F " A )  ~<_  A )
2019expcom 435 . 2  |-  ( A  e.  B  ->  (
( F " A
)  ~<_  dom  ( F  |`  A )  ->  ( F " A )  ~<_  A ) )
2112, 20syld 44 1  |-  ( A  e.  B  ->  ( Fun  F  ->  ( F " A )  ~<_  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   _Vcvv 2993    i^i cin 3348    C_ wss 3349   class class class wbr 4313   dom cdm 4861   ran crn 4862    |` cres 4863   "cima 4864   Fun wfun 5433   -onto->wfo 5437    ~<_ cdom 7329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-ac2 8653
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-recs 6853  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-card 8130  df-acn 8133  df-ac 8307
This theorem is referenced by:  uniimadom  8729  hausmapdom  19126  fimact  26039
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