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Theorem fodomfi2 8441
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 5797 . . . 4  |-  ( F : A -onto-> B  ->  F  Fn  A )
213ad2ant3 1019 . . 3  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  F  Fn  A )
3 forn 5798 . . . . 5  |-  ( F : A -onto-> B  ->  ran  F  =  B )
4 eqimss2 3557 . . . . 5  |-  ( ran 
F  =  B  ->  B  C_  ran  F )
53, 4syl 16 . . . 4  |-  ( F : A -onto-> B  ->  B  C_  ran  F )
653ad2ant3 1019 . . 3  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  B  C_ 
ran  F )
7 simp2 997 . . 3  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  B  e.  Fin )
8 fipreima 7826 . . 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 1228 . 2  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  E. x  e.  ( ~P A  i^i  Fin ) ( F "
x )  =  B )
10 inss2 3719 . . . . . . . . 9  |-  ( ~P A  i^i  Fin )  C_ 
Fin
1110sseli 3500 . . . . . . . 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 8329 . . . . . . 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 1001 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  F : A -onto-> B )
16 fofun 5796 . . . . . . . 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 3718 . . . . . . . . . . 11  |-  ( ~P A  i^i  Fin )  C_ 
~P A
1918sseli 3500 . . . . . . . . . 10  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
2019elpwid 4020 . . . . . . . . 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 5795 . . . . . . . . 9  |-  ( F : A -onto-> B  ->  F : A --> B )
23 fdm 5735 . . . . . . . . 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 3541 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  C_ 
dom  F )
26 fores 5804 . . . . . . 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 8438 . . . . . 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 999 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  A  e.  V )
31 ssdomg 7561 . . . . . 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 7568 . . . . 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 4450 . . . 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 2955 . 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 973    = wceq 1379    e. wcel 1767   E.wrex 2815    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   class class class wbr 4447   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002   Fun wfun 5582    Fn wfn 5583   -->wf 5584   -onto->wfo 5586    ~<_ cdom 7514   Fincfn 7516   cardccrd 8316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-1o 7130  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-fin 7520  df-card 8320  df-acn 8323
This theorem is referenced by:  wdomfil  8442
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