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Theorem fipreima 7412
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 959 . . 3  |-  ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  ->  A  e.  Fin )
2 dfss3 3338 . . . . . 6  |-  ( A 
C_  ran  F  <->  A. x  e.  A  x  e.  ran  F )
3 fvelrnb 5774 . . . . . . 7  |-  ( F  Fn  B  ->  (
x  e.  ran  F  <->  E. y  e.  B  ( F `  y )  =  x ) )
43ralbidv 2725 . . . . . 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 249 . . . . 5  |-  ( F  Fn  B  ->  ( A  C_  ran  F  <->  A. x  e.  A  E. y  e.  B  ( F `  y )  =  x ) )
65biimpa 471 . . . 4  |-  ( ( F  Fn  B  /\  A  C_  ran  F )  ->  A. x  e.  A  E. y  e.  B  ( F `  y )  =  x )
763adant3 977 . . 3  |-  ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  ->  A. x  e.  A  E. y  e.  B  ( F `  y )  =  x )
8 fveq2 5728 . . . . 5  |-  ( y  =  ( f `  x )  ->  ( F `  y )  =  ( F `  ( f `  x
) ) )
98eqeq1d 2444 . . . 4  |-  ( y  =  ( f `  x )  ->  (
( F `  y
)  =  x  <->  ( F `  ( f `  x
) )  =  x ) )
109ac6sfi 7351 . . 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 643 . 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 5216 . . . . . . 7  |-  ( f
" A )  C_  ran  f
13 frn 5597 . . . . . . 7  |-  ( f : A --> B  ->  ran  f  C_  B )
1412, 13syl5ss 3359 . . . . . 6  |-  ( f : A --> B  -> 
( f " A
)  C_  B )
15 vex 2959 . . . . . . . 8  |-  f  e. 
_V
16 imaexg 5217 . . . . . . . 8  |-  ( f  e.  _V  ->  (
f " A )  e.  _V )
1715, 16ax-mp 8 . . . . . . 7  |-  ( f
" A )  e. 
_V
1817elpw 3805 . . . . . 6  |-  ( ( f " A )  e.  ~P B  <->  ( f " A )  C_  B
)
1914, 18sylibr 204 . . . . 5  |-  ( f : A --> B  -> 
( f " A
)  e.  ~P B
)
2019ad2antrl 709 . . . 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 5593 . . . . . 6  |-  ( f : A --> B  ->  Fun  f )
2221ad2antrl 709 . . . . 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 962 . . . . 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 7399 . . . . 5  |-  ( ( Fun  f  /\  A  e.  Fin )  ->  (
f " A )  e.  Fin )
2522, 23, 24syl2anc 643 . . . 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 )
26 elin 3530 . . . 4  |-  ( ( f " A )  e.  ( ~P B  i^i  Fin )  <->  ( (
f " A )  e.  ~P B  /\  ( f " A
)  e.  Fin )
)
2720, 25, 26sylanbrc 646 . . 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 )
)
28 fvco3 5800 . . . . . . . . . . 11  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( ( F  o.  f ) `  x
)  =  ( F `
 ( f `  x ) ) )
29 fvresi 5924 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
3029adantl 453 . . . . . . . . . . 11  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( (  _I  |`  A ) `
 x )  =  x )
3128, 30eqeq12d 2450 . . . . . . . . . 10  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( ( ( F  o.  f ) `  x )  =  ( (  _I  |`  A ) `
 x )  <->  ( F `  ( f `  x
) )  =  x ) )
3231ralbidva 2721 . . . . . . . . 9  |-  ( f : A --> B  -> 
( A. x  e.  A  ( ( F  o.  f ) `  x )  =  ( (  _I  |`  A ) `
 x )  <->  A. x  e.  A  ( F `  ( f `  x
) )  =  x ) )
3332biimprd 215 . . . . . . . 8  |-  ( f : A --> B  -> 
( A. x  e.  A  ( F `  ( f `  x
) )  =  x  ->  A. x  e.  A  ( ( F  o.  f ) `  x
)  =  ( (  _I  |`  A ) `  x ) ) )
3433adantl 453 . . . . . . 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 ) ) )
3534impr 603 . . . . . 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 ) )
36 simpl1 960 . . . . . . . 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 )
37 ffn 5591 . . . . . . . . 9  |-  ( f : A --> B  -> 
f  Fn  A )
3837ad2antrl 709 . . . . . . . 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 )
3913ad2antrl 709 . . . . . . . 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 )
40 fnco 5553 . . . . . . . 8  |-  ( ( F  Fn  B  /\  f  Fn  A  /\  ran  f  C_  B )  ->  ( F  o.  f )  Fn  A
)
4136, 38, 39, 40syl3anc 1184 . . . . . . 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 )
42 fnresi 5562 . . . . . . 7  |-  (  _I  |`  A )  Fn  A
43 eqfnfv 5827 . . . . . . 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
) ) )
4441, 42, 43sylancl 644 . . . . . 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
) ) )
4535, 44mpbird 224 . . . . 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 ) )
4645imaeq1d 5202 . . . 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 ) )
47 imaco 5375 . . . 4  |-  ( ( F  o.  f )
" A )  =  ( F " (
f " A ) )
48 ssid 3367 . . . . 5  |-  A  C_  A
49 resiima 5220 . . . . 5  |-  ( A 
C_  A  ->  (
(  _I  |`  A )
" A )  =  A )
5048, 49ax-mp 8 . . . 4  |-  ( (  _I  |`  A ) " A )  =  A
5146, 47, 503eqtr3g 2491 . . 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 )
52 imaeq2 5199 . . . . 5  |-  ( c  =  ( f " A )  ->  ( F " c )  =  ( F " (
f " A ) ) )
5352eqeq1d 2444 . . . 4  |-  ( c  =  ( f " A )  ->  (
( F " c
)  =  A  <->  ( F " ( f " A
) )  =  A ) )
5453rspcev 3052 . . 3  |-  ( ( ( 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 643 . 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 )
5611, 55exlimddv 1648 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 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   _Vcvv 2956    i^i cin 3319    C_ wss 3320   ~Pcpw 3799    _I cid 4493   ran crn 4879    |` cres 4880   "cima 4881    o. ccom 4882   Fun wfun 5448    Fn wfn 5449   -->wf 5450   ` cfv 5454   Fincfn 7109
This theorem is referenced by:  fodomfi2  7941  cmpfi  17471  elrfirn  26749  lmhmfgsplit  27161  hbtlem6  27310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-er 6905  df-en 7110  df-dom 7111  df-fin 7113
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