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Theorem fodomfi2 8230
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 5622 . . . 4  |-  ( F : A -onto-> B  ->  F  Fn  A )
213ad2ant3 1011 . . 3  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  F  Fn  A )
3 forn 5623 . . . . 5  |-  ( F : A -onto-> B  ->  ran  F  =  B )
4 eqimss2 3409 . . . . 5  |-  ( ran 
F  =  B  ->  B  C_  ran  F )
53, 4syl 16 . . . 4  |-  ( F : A -onto-> B  ->  B  C_  ran  F )
653ad2ant3 1011 . . 3  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  B  C_ 
ran  F )
7 simp2 989 . . 3  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  B  e.  Fin )
8 fipreima 7617 . . 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 1218 . 2  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  E. x  e.  ( ~P A  i^i  Fin ) ( F "
x )  =  B )
10 inss2 3571 . . . . . . . . 9  |-  ( ~P A  i^i  Fin )  C_ 
Fin
1110sseli 3352 . . . . . . . 8  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  Fin )
1211adantl 466 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  Fin )
13 finnum 8118 . . . . . . 7  |-  ( x  e.  Fin  ->  x  e.  dom  card )
1412, 13syl 16 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  dom  card )
15 simpl3 993 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  F : A -onto-> B )
16 fofun 5621 . . . . . . . 8  |-  ( F : A -onto-> B  ->  Fun  F )
1715, 16syl 16 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  Fun  F )
18 inss1 3570 . . . . . . . . . . 11  |-  ( ~P A  i^i  Fin )  C_ 
~P A
1918sseli 3352 . . . . . . . . . 10  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
2019elpwid 3870 . . . . . . . . 9  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  C_  A )
2120adantl 466 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  C_  A )
22 fof 5620 . . . . . . . . 9  |-  ( F : A -onto-> B  ->  F : A --> B )
23 fdm 5563 . . . . . . . . 9  |-  ( F : A --> B  ->  dom  F  =  A )
2415, 22, 233syl 20 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  dom  F  =  A )
2521, 24sseqtr4d 3393 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  C_ 
dom  F )
26 fores 5629 . . . . . . 7  |-  ( ( Fun  F  /\  x  C_ 
dom  F )  -> 
( F  |`  x
) : x -onto-> ( F " x ) )
2717, 25, 26syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  x ) : x -onto-> ( F "
x ) )
28 fodomnum 8227 . . . . . 6  |-  ( x  e.  dom  card  ->  ( ( F  |`  x
) : x -onto-> ( F " x )  ->  ( F "
x )  ~<_  x ) )
2914, 27, 28sylc 60 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( F " x )  ~<_  x )
30 simpl1 991 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  A  e.  V )
31 ssdomg 7355 . . . . . 6  |-  ( A  e.  V  ->  (
x  C_  A  ->  x  ~<_  A ) )
3230, 21, 31sylc 60 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  ~<_  A )
33 domtr 7362 . . . . 5  |-  ( ( ( F " x
)  ~<_  x  /\  x  ~<_  A )  ->  ( F " x )  ~<_  A )
3429, 32, 33syl2anc 661 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( F " x )  ~<_  A )
35 breq1 4295 . . . 4  |-  ( ( F " x )  =  B  ->  (
( F " x
)  ~<_  A  <->  B  ~<_  A ) )
3634, 35syl5ibcom 220 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( F " x
)  =  B  ->  B  ~<_  A ) )
3736rexlimdva 2841 . 2  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  ( E. x  e.  ( ~P A  i^i  Fin )
( F " x
)  =  B  ->  B  ~<_  A ) )
389, 37mpd 15 1  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  B  ~<_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2716    i^i cin 3327    C_ wss 3328   ~Pcpw 3860   class class class wbr 4292   dom cdm 4840   ran crn 4841    |` cres 4842   "cima 4843   Fun wfun 5412    Fn wfn 5413   -->wf 5414   -onto->wfo 5416    ~<_ cdom 7308   Fincfn 7310   cardccrd 8105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-1o 6920  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-fin 7314  df-card 8109  df-acn 8112
This theorem is referenced by:  wdomfil  8231
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