MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fodomfi2 Unicode version

Theorem fodomfi2 7703
Description: Onto functions define dominance when a finite number of choices need to be made. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Assertion
Ref Expression
fodomfi2  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  B  ~<_  A )

Proof of Theorem fodomfi2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fofn 5469 . . . 4  |-  ( F : A -onto-> B  ->  F  Fn  A )
213ad2ant3 978 . . 3  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  F  Fn  A )
3 forn 5470 . . . . 5  |-  ( F : A -onto-> B  ->  ran  F  =  B )
4 eqimss2 3244 . . . . 5  |-  ( ran 
F  =  B  ->  B  C_  ran  F )
53, 4syl 15 . . . 4  |-  ( F : A -onto-> B  ->  B  C_  ran  F )
653ad2ant3 978 . . 3  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  B  C_ 
ran  F )
7 simp2 956 . . 3  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  B  e.  Fin )
8 fipreima 7177 . . 3  |-  ( ( F  Fn  A  /\  B  C_  ran  F  /\  B  e.  Fin )  ->  E. x  e.  ( ~P A  i^i  Fin ) ( F "
x )  =  B )
92, 6, 7, 8syl3anc 1182 . 2  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  E. x  e.  ( ~P A  i^i  Fin ) ( F "
x )  =  B )
10 inss2 3403 . . . . . . . . 9  |-  ( ~P A  i^i  Fin )  C_ 
Fin
1110sseli 3189 . . . . . . . 8  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  Fin )
1211adantl 452 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  Fin )
13 finnum 7597 . . . . . . 7  |-  ( x  e.  Fin  ->  x  e.  dom  card )
1412, 13syl 15 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  dom  card )
15 simpl3 960 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  F : A -onto-> B )
16 fofun 5468 . . . . . . . 8  |-  ( F : A -onto-> B  ->  Fun  F )
1715, 16syl 15 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  Fun  F )
18 inss1 3402 . . . . . . . . . . 11  |-  ( ~P A  i^i  Fin )  C_ 
~P A
1918sseli 3189 . . . . . . . . . 10  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
20 elpwi 3646 . . . . . . . . . 10  |-  ( x  e.  ~P A  ->  x  C_  A )
2119, 20syl 15 . . . . . . . . 9  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  C_  A )
2221adantl 452 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  C_  A )
23 fof 5467 . . . . . . . . 9  |-  ( F : A -onto-> B  ->  F : A --> B )
24 fdm 5409 . . . . . . . . 9  |-  ( F : A --> B  ->  dom  F  =  A )
2515, 23, 243syl 18 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  dom  F  =  A )
2622, 25sseqtr4d 3228 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  C_ 
dom  F )
27 fores 5476 . . . . . . 7  |-  ( ( Fun  F  /\  x  C_ 
dom  F )  -> 
( F  |`  x
) : x -onto-> ( F " x ) )
2817, 26, 27syl2anc 642 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  x ) : x -onto-> ( F "
x ) )
29 fodomnum 7700 . . . . . 6  |-  ( x  e.  dom  card  ->  ( ( F  |`  x
) : x -onto-> ( F " x )  ->  ( F "
x )  ~<_  x ) )
3014, 28, 29sylc 56 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( F " x )  ~<_  x )
31 simpl1 958 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  A  e.  V )
32 ssdomg 6923 . . . . . 6  |-  ( A  e.  V  ->  (
x  C_  A  ->  x  ~<_  A ) )
3331, 22, 32sylc 56 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  ~<_  A )
34 domtr 6930 . . . . 5  |-  ( ( ( F " x
)  ~<_  x  /\  x  ~<_  A )  ->  ( F " x )  ~<_  A )
3530, 33, 34syl2anc 642 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( F " x )  ~<_  A )
36 breq1 4042 . . . 4  |-  ( ( F " x )  =  B  ->  (
( F " x
)  ~<_  A  <->  B  ~<_  A ) )
3735, 36syl5ibcom 211 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( F " x
)  =  B  ->  B  ~<_  A ) )
3837rexlimdva 2680 . 2  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  ( E. x  e.  ( ~P A  i^i  Fin )
( F " x
)  =  B  ->  B  ~<_  A ) )
399, 38mpd 14 1  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  B  ~<_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   class class class wbr 4039   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265    Fn wfn 5266   -->wf 5267   -onto->wfo 5269    ~<_ cdom 6877   Fincfn 6879   cardccrd 7584
This theorem is referenced by:  wdomfil  7704
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-rep 4147  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-rmo 2564  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-int 3879  df-iun 3923  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-se 4369  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-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-1o 6495  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-fin 6883  df-card 7588  df-acn 7591
  Copyright terms: Public domain W3C validator