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Theorem fipreima 7161
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 957 . . 3  |-  ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  ->  A  e.  Fin )
2 dfss3 3170 . . . . . 6  |-  ( A 
C_  ran  F  <->  A. x  e.  A  x  e.  ran  F )
3 fvelrnb 5570 . . . . . . 7  |-  ( F  Fn  B  ->  (
x  e.  ran  F  <->  E. y  e.  B  ( F `  y )  =  x ) )
43ralbidv 2563 . . . . . 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 248 . . . . 5  |-  ( F  Fn  B  ->  ( A  C_  ran  F  <->  A. x  e.  A  E. y  e.  B  ( F `  y )  =  x ) )
65biimpa 470 . . . 4  |-  ( ( F  Fn  B  /\  A  C_  ran  F )  ->  A. x  e.  A  E. y  e.  B  ( F `  y )  =  x )
763adant3 975 . . 3  |-  ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  ->  A. x  e.  A  E. y  e.  B  ( F `  y )  =  x )
8 fveq2 5525 . . . . 5  |-  ( y  =  ( f `  x )  ->  ( F `  y )  =  ( F `  ( f `  x
) ) )
98eqeq1d 2291 . . . 4  |-  ( y  =  ( f `  x )  ->  (
( F `  y
)  =  x  <->  ( F `  ( f `  x
) )  =  x ) )
109ac6sfi 7101 . . 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 642 . 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 5025 . . . . . . . . 9  |-  ( f
" A )  C_  ran  f
13 frn 5395 . . . . . . . . 9  |-  ( f : A --> B  ->  ran  f  C_  B )
1412, 13syl5ss 3190 . . . . . . . 8  |-  ( f : A --> B  -> 
( f " A
)  C_  B )
15 vex 2791 . . . . . . . . . 10  |-  f  e. 
_V
16 imaexg 5026 . . . . . . . . . 10  |-  ( f  e.  _V  ->  (
f " A )  e.  _V )
1715, 16ax-mp 8 . . . . . . . . 9  |-  ( f
" A )  e. 
_V
1817elpw 3631 . . . . . . . 8  |-  ( ( f " A )  e.  ~P B  <->  ( f " A )  C_  B
)
1914, 18sylibr 203 . . . . . . 7  |-  ( f : A --> B  -> 
( f " A
)  e.  ~P B
)
2019ad2antrl 708 . . . . . 6  |-  ( ( ( 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 5391 . . . . . . . 8  |-  ( f : A --> B  ->  Fun  f )
2221ad2antrl 708 . . . . . . 7  |-  ( ( ( 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 960 . . . . . . 7  |-  ( ( ( 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 7148 . . . . . . 7  |-  ( ( Fun  f  /\  A  e.  Fin )  ->  (
f " A )  e.  Fin )
2522, 23, 24syl2anc 642 . . . . . 6  |-  ( ( ( 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 )
26 elin 3358 . . . . . 6  |-  ( ( f " A )  e.  ( ~P B  i^i  Fin )  <->  ( (
f " A )  e.  ~P B  /\  ( f " A
)  e.  Fin )
)
2720, 25, 26sylanbrc 645 . . . . 5  |-  ( ( ( 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 )
)
28 fvco3 5596 . . . . . . . . . . . . 13  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( ( F  o.  f ) `  x
)  =  ( F `
 ( f `  x ) ) )
29 fvresi 5711 . . . . . . . . . . . . . 14  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
3029adantl 452 . . . . . . . . . . . . 13  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( (  _I  |`  A ) `
 x )  =  x )
3128, 30eqeq12d 2297 . . . . . . . . . . . 12  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( ( ( F  o.  f ) `  x )  =  ( (  _I  |`  A ) `
 x )  <->  ( F `  ( f `  x
) )  =  x ) )
3231ralbidva 2559 . . . . . . . . . . 11  |-  ( f : A --> B  -> 
( A. x  e.  A  ( ( F  o.  f ) `  x )  =  ( (  _I  |`  A ) `
 x )  <->  A. x  e.  A  ( F `  ( f `  x
) )  =  x ) )
3332biimprd 214 . . . . . . . . . 10  |-  ( f : A --> B  -> 
( A. x  e.  A  ( F `  ( f `  x
) )  =  x  ->  A. x  e.  A  ( ( F  o.  f ) `  x
)  =  ( (  _I  |`  A ) `  x ) ) )
3433adantl 452 . . . . . . . . 9  |-  ( ( ( 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 ) ) )
3534impr 602 . . . . . . . 8  |-  ( ( ( 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 ) )
36 simpl1 958 . . . . . . . . . 10  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  ->  F  Fn  B )
37 ffn 5389 . . . . . . . . . . 11  |-  ( f : A --> B  -> 
f  Fn  A )
3837ad2antrl 708 . . . . . . . . . 10  |-  ( ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  /\  ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `  x ) )  =  x ) )  -> 
f  Fn  A )
3913ad2antrl 708 . . . . . . . . . 10  |-  ( ( ( 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 )
40 fnco 5352 . . . . . . . . . 10  |-  ( ( F  Fn  B  /\  f  Fn  A  /\  ran  f  C_  B )  ->  ( F  o.  f )  Fn  A
)
4136, 38, 39, 40syl3anc 1182 . . . . . . . . 9  |-  ( ( ( 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 )
42 fnresi 5361 . . . . . . . . 9  |-  (  _I  |`  A )  Fn  A
43 eqfnfv 5622 . . . . . . . . 9  |-  ( ( ( 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
) ) )
4441, 42, 43sylancl 643 . . . . . . . 8  |-  ( ( ( 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
) ) )
4535, 44mpbird 223 . . . . . . 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
)  =  (  _I  |`  A ) )
4645imaeq1d 5011 . . . . . 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 ) " A
)  =  ( (  _I  |`  A ) " A ) )
47 imaco 5178 . . . . . 6  |-  ( ( F  o.  f )
" A )  =  ( F " (
f " A ) )
48 ssid 3197 . . . . . . 7  |-  A  C_  A
49 resiima 5029 . . . . . . 7  |-  ( A 
C_  A  ->  (
(  _I  |`  A )
" A )  =  A )
5048, 49ax-mp 8 . . . . . 6  |-  ( (  _I  |`  A ) " A )  =  A
5146, 47, 503eqtr3g 2338 . . . . 5  |-  ( ( ( 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 )
52 imaeq2 5008 . . . . . . 7  |-  ( c  =  ( f " A )  ->  ( F " c )  =  ( F " (
f " A ) ) )
5352eqeq1d 2291 . . . . . 6  |-  ( c  =  ( f " A )  ->  (
( F " c
)  =  A  <->  ( F " ( f " A
) )  =  A ) )
5453rspcev 2884 . . . . 5  |-  ( ( ( f " A
)  e.  ( ~P B  i^i  Fin )  /\  ( F " (
f " A ) )  =  A )  ->  E. c  e.  ( ~P B  i^i  Fin ) ( F "
c )  =  A )
5527, 51, 54syl2anc 642 . . . 4  |-  ( ( ( 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 )
5655ex 423 . . 3  |-  ( ( 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 ) )
5756exlimdv 1664 . 2  |-  ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  ->  ( E. f ( f : A --> B  /\  A. x  e.  A  ( F `  ( f `
 x ) )  =  x )  ->  E. c  e.  ( ~P B  i^i  Fin )
( F " c
)  =  A ) )
5811, 57mpd 14 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625    _I cid 4304   ran crn 4690    |` cres 4691   "cima 4692    o. ccom 4693   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255   Fincfn 6863
This theorem is referenced by:  fodomfi2  7687  cmpfi  17135  fipreimaOLD  26415  elrfirn  26770  lmhmfgsplit  27184  hbtlem6  27333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-fin 6867
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