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Theorem fidomdm 7871
Description: Any finite set dominates its domain. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
fidomdm  |-  ( F  e.  Fin  ->  dom  F  ~<_  F )

Proof of Theorem fidomdm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dmresv 5301 . 2  |-  dom  ( F  |`  _V )  =  dom  F
2 finresfin 7815 . . . 4  |-  ( F  e.  Fin  ->  ( F  |`  _V )  e. 
Fin )
3 fvex 5889 . . . . . . 7  |-  ( 1st `  x )  e.  _V
4 eqid 2471 . . . . . . 7  |-  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) )  =  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x
) )
53, 4fnmpti 5716 . . . . . 6  |-  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) )  Fn  ( F  |`  _V )
6 dffn4 5812 . . . . . 6  |-  ( ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) )  Fn  ( F  |`  _V )  <->  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) ) : ( F  |`  _V ) -onto-> ran  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x
) ) )
75, 6mpbi 213 . . . . 5  |-  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) ) : ( F  |`  _V ) -onto-> ran  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x
) )
8 relres 5138 . . . . . 6  |-  Rel  ( F  |`  _V )
9 reldm 6863 . . . . . 6  |-  ( Rel  ( F  |`  _V )  ->  dom  ( F  |`  _V )  =  ran  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) ) )
10 foeq3 5804 . . . . . 6  |-  ( dom  ( F  |`  _V )  =  ran  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x
) )  ->  (
( x  e.  ( F  |`  _V )  |->  ( 1st `  x
) ) : ( F  |`  _V ) -onto-> dom  ( F  |`  _V )  <->  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) ) : ( F  |`  _V ) -onto-> ran  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x
) ) ) )
118, 9, 10mp2b 10 . . . . 5  |-  ( ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) ) : ( F  |`  _V ) -onto-> dom  ( F  |`  _V )  <->  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) ) : ( F  |`  _V ) -onto-> ran  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x
) ) )
127, 11mpbir 214 . . . 4  |-  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) ) : ( F  |`  _V ) -onto-> dom  ( F  |`  _V )
13 fodomfi 7868 . . . 4  |-  ( ( ( F  |`  _V )  e.  Fin  /\  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) ) : ( F  |`  _V ) -onto-> dom  ( F  |`  _V )
)  ->  dom  ( F  |`  _V )  ~<_  ( F  |`  _V ) )
142, 12, 13sylancl 675 . . 3  |-  ( F  e.  Fin  ->  dom  ( F  |`  _V )  ~<_  ( F  |`  _V )
)
15 resss 5134 . . . 4  |-  ( F  |`  _V )  C_  F
16 ssdomg 7633 . . . 4  |-  ( F  e.  Fin  ->  (
( F  |`  _V )  C_  F  ->  ( F  |` 
_V )  ~<_  F ) )
1715, 16mpi 20 . . 3  |-  ( F  e.  Fin  ->  ( F  |`  _V )  ~<_  F )
18 domtr 7640 . . 3  |-  ( ( dom  ( F  |`  _V )  ~<_  ( F  |` 
_V )  /\  ( F  |`  _V )  ~<_  F )  ->  dom  ( F  |`  _V )  ~<_  F )
1914, 17, 18syl2anc 673 . 2  |-  ( F  e.  Fin  ->  dom  ( F  |`  _V )  ~<_  F )
201, 19syl5eqbrr 4430 1  |-  ( F  e.  Fin  ->  dom  F  ~<_  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1452    e. wcel 1904   _Vcvv 3031    C_ wss 3390   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   ran crn 4840    |` cres 4841   Rel wrel 4844    Fn wfn 5584   -onto->wfo 5587   ` cfv 5589   1stc1st 6810    ~<_ cdom 7585   Fincfn 7587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-1st 6812  df-2nd 6813  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-fin 7591
This theorem is referenced by:  dmfi  7872  hashfun  12650
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