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Theorem imadomg 8903
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 5001 . . . 4  |-  ( F
" A )  =  ran  ( F  |`  A )
2 resfunexg 6112 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F  |`  A )  e. 
_V )
3 dmexg 6704 . . . . . 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 5609 . . . . . . 7  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
6 funforn 5784 . . . . . . 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 463 . . . . 5  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F  |`  A ) : dom  ( F  |`  A ) -onto-> ran  ( F  |`  A ) )
9 fodomg 8894 . . . . 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 4472 . . 3  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F " A )  ~<_  dom  ( F  |`  A ) )
1211expcom 433 . 2  |-  ( A  e.  B  ->  ( Fun  F  ->  ( F " A )  ~<_  dom  ( F  |`  A ) ) )
13 dmres 5282 . . . . . 6  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
14 inss1 3704 . . . . . 6  |-  ( A  i^i  dom  F )  C_  A
1513, 14eqsstri 3519 . . . . 5  |-  dom  ( F  |`  A )  C_  A
16 ssdomg 7554 . . . . 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 7561 . . . 4  |-  ( ( ( F " A
)  ~<_  dom  ( F  |`  A )  /\  dom  ( F  |`  A )  ~<_  A )  ->  ( F " A )  ~<_  A )
1917, 18sylan2 472 . . 3  |-  ( ( ( F " A
)  ~<_  dom  ( F  |`  A )  /\  A  e.  B )  ->  ( F " A )  ~<_  A )
2019expcom 433 . 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 367    e. wcel 1823   _Vcvv 3106    i^i cin 3460    C_ wss 3461   class class class wbr 4439   dom cdm 4988   ran crn 4989    |` cres 4990   "cima 4991   Fun wfun 5564   -onto->wfo 5568    ~<_ cdom 7507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-ac2 8834
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-recs 7034  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-card 8311  df-acn 8314  df-ac 8488
This theorem is referenced by:  uniimadom  8910  hausmapdom  20170  fimact  27773
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