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Theorem fipreima 7177
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 3183 . . . . . 6  |-  ( A 
C_  ran  F  <->  A. x  e.  A  x  e.  ran  F )
3 fvelrnb 5586 . . . . . . 7  |-  ( F  Fn  B  ->  (
x  e.  ran  F  <->  E. y  e.  B  ( F `  y )  =  x ) )
43ralbidv 2576 . . . . . 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 5541 . . . . 5  |-  ( y  =  ( f `  x )  ->  ( F `  y )  =  ( F `  ( f `  x
) ) )
98eqeq1d 2304 . . . 4  |-  ( y  =  ( f `  x )  ->  (
( F `  y
)  =  x  <->  ( F `  ( f `  x
) )  =  x ) )
109ac6sfi 7117 . . 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 5041 . . . . . . . . 9  |-  ( f
" A )  C_  ran  f
13 frn 5411 . . . . . . . . 9  |-  ( f : A --> B  ->  ran  f  C_  B )
1412, 13syl5ss 3203 . . . . . . . 8  |-  ( f : A --> B  -> 
( f " A
)  C_  B )
15 vex 2804 . . . . . . . . . 10  |-  f  e. 
_V
16 imaexg 5042 . . . . . . . . . 10  |-  ( f  e.  _V  ->  (
f " A )  e.  _V )
1715, 16ax-mp 8 . . . . . . . . 9  |-  ( f
" A )  e. 
_V
1817elpw 3644 . . . . . . . 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 5407 . . . . . . . 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 7164 . . . . . . 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 3371 . . . . . 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 5612 . . . . . . . . . . . . 13  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( ( F  o.  f ) `  x
)  =  ( F `
 ( f `  x ) ) )
29 fvresi 5727 . . . . . . . . . . . . . 14  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
3029adantl 452 . . . . . . . . . . . . 13  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( (  _I  |`  A ) `
 x )  =  x )
3128, 30eqeq12d 2310 . . . . . . . . . . . 12  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( ( ( F  o.  f ) `  x )  =  ( (  _I  |`  A ) `
 x )  <->  ( F `  ( f `  x
) )  =  x ) )
3231ralbidva 2572 . . . . . . . . . . 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 5405 . . . . . . . . . . 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 5368 . . . . . . . . . 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 5377 . . . . . . . . 9  |-  (  _I  |`  A )  Fn  A
43 eqfnfv 5638 . . . . . . . . 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 5027 . . . . . 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 5194 . . . . . 6  |-  ( ( F  o.  f )
" A )  =  ( F " (
f " A ) )
48 ssid 3210 . . . . . . 7  |-  A  C_  A
49 resiima 5045 . . . . . . 7  |-  ( A 
C_  A  ->  (
(  _I  |`  A )
" A )  =  A )
5048, 49ax-mp 8 . . . . . 6  |-  ( (  _I  |`  A ) " A )  =  A
5146, 47, 503eqtr3g 2351 . . . . 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 5024 . . . . . . 7  |-  ( c  =  ( f " A )  ->  ( F " c )  =  ( F " (
f " A ) ) )
5352eqeq1d 2304 . . . . . 6  |-  ( c  =  ( f " A )  ->  (
( F " c
)  =  A  <->  ( F " ( f " A
) )  =  A ) )
5453rspcev 2897 . . . . 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 1626 . 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 1531    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ~Pcpw 3638    _I cid 4320   ran crn 4706    |` cres 4707   "cima 4708    o. ccom 4709   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271   Fincfn 6879
This theorem is referenced by:  fodomfi2  7703  cmpfi  17151  fipreimaOLD  26518  elrfirn  26873  lmhmfgsplit  27287  hbtlem6  27436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-fin 6883
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