MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imadomg Structured version   Unicode version

Theorem imadomg 8912
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 5012 . . . 4  |-  ( F
" A )  =  ran  ( F  |`  A )
2 resfunexg 6126 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F  |`  A )  e. 
_V )
3 dmexg 6715 . . . . . 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 5627 . . . . . . 7  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
6 funforn 5802 . . . . . . 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 8903 . . . . 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 4480 . . 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 5294 . . . . . 6  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
14 inss1 3718 . . . . . 6  |-  ( A  i^i  dom  F )  C_  A
1513, 14eqsstri 3534 . . . . 5  |-  dom  ( F  |`  A )  C_  A
16 ssdomg 7561 . . . . 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 7568 . . . 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 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476   class class class wbr 4447   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002   Fun wfun 5582   -onto->wfo 5586    ~<_ cdom 7514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-ac2 8843
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-recs 7042  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-card 8320  df-acn 8323  df-ac 8497
This theorem is referenced by:  uniimadom  8919  hausmapdom  19795  fimact  27240
  Copyright terms: Public domain W3C validator