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Theorem fodomfi2 8226
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 5619 . . . 4  |-  ( F : A -onto-> B  ->  F  Fn  A )
213ad2ant3 1006 . . 3  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  F  Fn  A )
3 forn 5620 . . . . 5  |-  ( F : A -onto-> B  ->  ran  F  =  B )
4 eqimss2 3406 . . . . 5  |-  ( ran 
F  =  B  ->  B  C_  ran  F )
53, 4syl 16 . . . 4  |-  ( F : A -onto-> B  ->  B  C_  ran  F )
653ad2ant3 1006 . . 3  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  B  C_ 
ran  F )
7 simp2 984 . . 3  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  B  e.  Fin )
8 fipreima 7613 . . 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 1213 . 2  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  E. x  e.  ( ~P A  i^i  Fin ) ( F "
x )  =  B )
10 inss2 3568 . . . . . . . . 9  |-  ( ~P A  i^i  Fin )  C_ 
Fin
1110sseli 3349 . . . . . . . 8  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  Fin )
1211adantl 463 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  Fin )
13 finnum 8114 . . . . . . 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 988 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  F : A -onto-> B )
16 fofun 5618 . . . . . . . 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 3567 . . . . . . . . . . 11  |-  ( ~P A  i^i  Fin )  C_ 
~P A
1918sseli 3349 . . . . . . . . . 10  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
2019elpwid 3867 . . . . . . . . 9  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  C_  A )
2120adantl 463 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  C_  A )
22 fof 5617 . . . . . . . . 9  |-  ( F : A -onto-> B  ->  F : A --> B )
23 fdm 5560 . . . . . . . . 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 3390 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  C_ 
dom  F )
26 fores 5626 . . . . . . 7  |-  ( ( Fun  F  /\  x  C_ 
dom  F )  -> 
( F  |`  x
) : x -onto-> ( F " x ) )
2717, 25, 26syl2anc 656 . . . . . 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 8223 . . . . . 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 986 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  A  e.  V )
31 ssdomg 7351 . . . . . 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 7358 . . . . 5  |-  ( ( ( F " x
)  ~<_  x  /\  x  ~<_  A )  ->  ( F " x )  ~<_  A )
3429, 32, 33syl2anc 656 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( F " x )  ~<_  A )
35 breq1 4292 . . . 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 2839 . 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 960    = wceq 1364    e. wcel 1761   E.wrex 2714    i^i cin 3324    C_ wss 3325   ~Pcpw 3857   class class class wbr 4289   dom cdm 4836   ran crn 4837    |` cres 4838   "cima 4839   Fun wfun 5409    Fn wfn 5410   -->wf 5411   -onto->wfo 5413    ~<_ cdom 7304   Fincfn 7306   cardccrd 8101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-1o 6916  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-fin 7310  df-card 8105  df-acn 8108
This theorem is referenced by:  wdomfil  8227
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