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Theorem fipreima 7370
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 3298 . . . . . 6  |-  ( A 
C_  ran  F  <->  A. x  e.  A  x  e.  ran  F )
3 fvelrnb 5733 . . . . . . 7  |-  ( F  Fn  B  ->  (
x  e.  ran  F  <->  E. y  e.  B  ( F `  y )  =  x ) )
43ralbidv 2686 . . . . . 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 5687 . . . . 5  |-  ( y  =  ( f `  x )  ->  ( F `  y )  =  ( F `  ( f `  x
) ) )
98eqeq1d 2412 . . . 4  |-  ( y  =  ( f `  x )  ->  (
( F `  y
)  =  x  <->  ( F `  ( f `  x
) )  =  x ) )
109ac6sfi 7310 . . 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 5175 . . . . . . 7  |-  ( f
" A )  C_  ran  f
13 frn 5556 . . . . . . 7  |-  ( f : A --> B  ->  ran  f  C_  B )
1412, 13syl5ss 3319 . . . . . 6  |-  ( f : A --> B  -> 
( f " A
)  C_  B )
15 vex 2919 . . . . . . . 8  |-  f  e. 
_V
16 imaexg 5176 . . . . . . . 8  |-  ( f  e.  _V  ->  (
f " A )  e.  _V )
1715, 16ax-mp 8 . . . . . . 7  |-  ( f
" A )  e. 
_V
1817elpw 3765 . . . . . 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 5552 . . . . . 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 7357 . . . . 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 3490 . . . 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 5759 . . . . . . . . . . 11  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( ( F  o.  f ) `  x
)  =  ( F `
 ( f `  x ) ) )
29 fvresi 5883 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
3029adantl 453 . . . . . . . . . . 11  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( (  _I  |`  A ) `
 x )  =  x )
3128, 30eqeq12d 2418 . . . . . . . . . 10  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( ( ( F  o.  f ) `  x )  =  ( (  _I  |`  A ) `
 x )  <->  ( F `  ( f `  x
) )  =  x ) )
3231ralbidva 2682 . . . . . . . . 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 5550 . . . . . . . . 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 5512 . . . . . . . 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 5521 . . . . . . 7  |-  (  _I  |`  A )  Fn  A
43 eqfnfv 5786 . . . . . . 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 5161 . . . 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 5334 . . . 4  |-  ( ( F  o.  f )
" A )  =  ( F " (
f " A ) )
48 ssid 3327 . . . . 5  |-  A  C_  A
49 resiima 5179 . . . . 5  |-  ( A 
C_  A  ->  (
(  _I  |`  A )
" A )  =  A )
5048, 49ax-mp 8 . . . 4  |-  ( (  _I  |`  A ) " A )  =  A
5146, 47, 503eqtr3g 2459 . . 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 5158 . . . . 5  |-  ( c  =  ( f " A )  ->  ( F " c )  =  ( F " (
f " A ) ) )
5352eqeq1d 2412 . . . 4  |-  ( c  =  ( f " A )  ->  (
( F " c
)  =  A  <->  ( F " ( f " A
) )  =  A ) )
5453rspcev 3012 . . 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 1645 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 1547    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916    i^i cin 3279    C_ wss 3280   ~Pcpw 3759    _I cid 4453   ran crn 4838    |` cres 4839   "cima 4840    o. ccom 4841   Fun wfun 5407    Fn wfn 5408   -->wf 5409   ` cfv 5413   Fincfn 7068
This theorem is referenced by:  fodomfi2  7897  cmpfi  17425  elrfirn  26639  lmhmfgsplit  27052  hbtlem6  27201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-fin 7072
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