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Theorem fipreima 7838
Description: Given a finite subset  A of the range of a function, there exists a finite subset of the domain whose image is  A. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
fipreima  |-  ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  ->  E. c  e.  ( ~P B  i^i  Fin ) ( F "
c )  =  A )
Distinct variable groups:    A, c    B, c    F, c

Proof of Theorem fipreima
Dummy variables  f  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 998 . . 3  |-  ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  ->  A  e.  Fin )
2 dfss3 3499 . . . . . 6  |-  ( A 
C_  ran  F  <->  A. x  e.  A  x  e.  ran  F )
3 fvelrnb 5921 . . . . . . 7  |-  ( F  Fn  B  ->  (
x  e.  ran  F  <->  E. y  e.  B  ( F `  y )  =  x ) )
43ralbidv 2906 . . . . . 6  |-  ( F  Fn  B  ->  ( A. x  e.  A  x  e.  ran  F  <->  A. x  e.  A  E. y  e.  B  ( F `  y )  =  x ) )
52, 4syl5bb 257 . . . . 5  |-  ( F  Fn  B  ->  ( A  C_  ran  F  <->  A. x  e.  A  E. y  e.  B  ( F `  y )  =  x ) )
65biimpa 484 . . . 4  |-  ( ( F  Fn  B  /\  A  C_  ran  F )  ->  A. x  e.  A  E. y  e.  B  ( F `  y )  =  x )
763adant3 1016 . . 3  |-  ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  ->  A. x  e.  A  E. y  e.  B  ( F `  y )  =  x )
8 fveq2 5872 . . . . 5  |-  ( y  =  ( f `  x )  ->  ( F `  y )  =  ( F `  ( f `  x
) ) )
98eqeq1d 2469 . . . 4  |-  ( y  =  ( f `  x )  ->  (
( F `  y
)  =  x  <->  ( F `  ( f `  x
) )  =  x ) )
109ac6sfi 7776 . . 3  |-  ( ( A  e.  Fin  /\  A. x  e.  A  E. y  e.  B  ( F `  y )  =  x )  ->  E. f
( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x
) )  =  x ) )
111, 7, 10syl2anc 661 . 2  |-  ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  ->  E. f ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `
 x ) )  =  x ) )
12 imassrn 5354 . . . . . . 7  |-  ( f
" A )  C_  ran  f
13 frn 5743 . . . . . . 7  |-  ( f : A --> B  ->  ran  f  C_  B )
1412, 13syl5ss 3520 . . . . . 6  |-  ( f : A --> B  -> 
( f " A
)  C_  B )
15 vex 3121 . . . . . . . 8  |-  f  e. 
_V
16 imaexg 6732 . . . . . . . 8  |-  ( f  e.  _V  ->  (
f " A )  e.  _V )
1715, 16ax-mp 5 . . . . . . 7  |-  ( f
" A )  e. 
_V
1817elpw 4022 . . . . . 6  |-  ( ( f " A )  e.  ~P B  <->  ( f " A )  C_  B
)
1914, 18sylibr 212 . . . . 5  |-  ( f : A --> B  -> 
( f " A
)  e.  ~P B
)
2019ad2antrl 727 . . . 4  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  -> 
( f " A
)  e.  ~P B
)
21 ffun 5739 . . . . . 6  |-  ( f : A --> B  ->  Fun  f )
2221ad2antrl 727 . . . . 5  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  ->  Fun  f )
23 simpl3 1001 . . . . 5  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  ->  A  e.  Fin )
24 imafi 7825 . . . . 5  |-  ( ( Fun  f  /\  A  e.  Fin )  ->  (
f " A )  e.  Fin )
2522, 23, 24syl2anc 661 . . . 4  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  -> 
( f " A
)  e.  Fin )
2620, 25elind 3693 . . 3  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  -> 
( f " A
)  e.  ( ~P B  i^i  Fin )
)
27 fvco3 5951 . . . . . . . . . . 11  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( ( F  o.  f ) `  x
)  =  ( F `
 ( f `  x ) ) )
28 fvresi 6098 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
2928adantl 466 . . . . . . . . . . 11  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( (  _I  |`  A ) `
 x )  =  x )
3027, 29eqeq12d 2489 . . . . . . . . . 10  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( ( ( F  o.  f ) `  x )  =  ( (  _I  |`  A ) `
 x )  <->  ( F `  ( f `  x
) )  =  x ) )
3130ralbidva 2903 . . . . . . . . 9  |-  ( f : A --> B  -> 
( A. x  e.  A  ( ( F  o.  f ) `  x )  =  ( (  _I  |`  A ) `
 x )  <->  A. x  e.  A  ( F `  ( f `  x
) )  =  x ) )
3231biimprd 223 . . . . . . . 8  |-  ( f : A --> B  -> 
( A. x  e.  A  ( F `  ( f `  x
) )  =  x  ->  A. x  e.  A  ( ( F  o.  f ) `  x
)  =  ( (  _I  |`  A ) `  x ) ) )
3332adantl 466 . . . . . . 7  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  f : A --> B )  ->  ( A. x  e.  A  ( F `  ( f `
 x ) )  =  x  ->  A. x  e.  A  ( ( F  o.  f ) `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
3433impr 619 . . . . . 6  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  ->  A. x  e.  A  ( ( F  o.  f ) `  x
)  =  ( (  _I  |`  A ) `  x ) )
35 simpl1 999 . . . . . . . 8  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  ->  F  Fn  B )
36 ffn 5737 . . . . . . . . 9  |-  ( f : A --> B  -> 
f  Fn  A )
3736ad2antrl 727 . . . . . . . 8  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  -> 
f  Fn  A )
3813ad2antrl 727 . . . . . . . 8  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  ->  ran  f  C_  B )
39 fnco 5695 . . . . . . . 8  |-  ( ( F  Fn  B  /\  f  Fn  A  /\  ran  f  C_  B )  ->  ( F  o.  f )  Fn  A
)
4035, 37, 38, 39syl3anc 1228 . . . . . . 7  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  -> 
( F  o.  f
)  Fn  A )
41 fnresi 5704 . . . . . . 7  |-  (  _I  |`  A )  Fn  A
42 eqfnfv 5982 . . . . . . 7  |-  ( ( ( F  o.  f
)  Fn  A  /\  (  _I  |`  A )  Fn  A )  -> 
( ( F  o.  f )  =  (  _I  |`  A )  <->  A. x  e.  A  ( ( F  o.  f
) `  x )  =  ( (  _I  |`  A ) `  x
) ) )
4340, 41, 42sylancl 662 . . . . . 6  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  -> 
( ( F  o.  f )  =  (  _I  |`  A )  <->  A. x  e.  A  ( ( F  o.  f
) `  x )  =  ( (  _I  |`  A ) `  x
) ) )
4434, 43mpbird 232 . . . . 5  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  -> 
( F  o.  f
)  =  (  _I  |`  A ) )
4544imaeq1d 5342 . . . 4  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  -> 
( ( F  o.  f ) " A
)  =  ( (  _I  |`  A ) " A ) )
46 imaco 5518 . . . 4  |-  ( ( F  o.  f )
" A )  =  ( F " (
f " A ) )
47 ssid 3528 . . . . 5  |-  A  C_  A
48 resiima 5357 . . . . 5  |-  ( A 
C_  A  ->  (
(  _I  |`  A )
" A )  =  A )
4947, 48ax-mp 5 . . . 4  |-  ( (  _I  |`  A ) " A )  =  A
5045, 46, 493eqtr3g 2531 . . 3  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  -> 
( F " (
f " A ) )  =  A )
51 imaeq2 5339 . . . . 5  |-  ( c  =  ( f " A )  ->  ( F " c )  =  ( F " (
f " A ) ) )
5251eqeq1d 2469 . . . 4  |-  ( c  =  ( f " A )  ->  (
( F " c
)  =  A  <->  ( F " ( f " A
) )  =  A ) )
5352rspcev 3219 . . 3  |-  ( ( ( f " A
)  e.  ( ~P B  i^i  Fin )  /\  ( F " (
f " A ) )  =  A )  ->  E. c  e.  ( ~P B  i^i  Fin ) ( F "
c )  =  A )
5426, 50, 53syl2anc 661 . 2  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  ->  E. c  e.  ( ~P B  i^i  Fin )
( F " c
)  =  A )
5511, 54exlimddv 1702 1  |-  ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  ->  E. c  e.  ( ~P B  i^i  Fin ) ( F "
c )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   A.wral 2817   E.wrex 2818   _Vcvv 3118    i^i cin 3480    C_ wss 3481   ~Pcpw 4016    _I cid 4796   ran crn 5006    |` cres 5007   "cima 5008    o. ccom 5009   Fun wfun 5588    Fn wfn 5589   -->wf 5590   ` cfv 5594   Fincfn 7528
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-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-1o 7142  df-er 7323  df-en 7529  df-dom 7530  df-fin 7532
This theorem is referenced by:  fodomfi2  8453  cmpfi  19776  elrfirn  30555  lmhmfgsplit  30960  hbtlem6  31006
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