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Theorem fidomdm 7390
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 5331 . 2  |-  dom  ( F  |`  _V )  =  dom  F
2 finresfin 7336 . . . 4  |-  ( F  e.  Fin  ->  ( F  |`  _V )  e. 
Fin )
3 fvex 5744 . . . . . . 7  |-  ( 1st `  x )  e.  _V
4 eqid 2438 . . . . . . 7  |-  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) )  =  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x
) )
53, 4fnmpti 5575 . . . . . 6  |-  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) )  Fn  ( F  |`  _V )
6 dffn4 5661 . . . . . 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 201 . . . . 5  |-  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) ) : ( F  |`  _V ) -onto-> ran  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x
) )
8 relres 5176 . . . . . 6  |-  Rel  ( F  |`  _V )
9 reldm 6400 . . . . . 6  |-  ( Rel  ( F  |`  _V )  ->  dom  ( F  |`  _V )  =  ran  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) ) )
10 foeq3 5653 . . . . . 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 202 . . . 4  |-  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) ) : ( F  |`  _V ) -onto-> dom  ( F  |`  _V )
13 fodomfi 7387 . . . 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 645 . . 3  |-  ( F  e.  Fin  ->  dom  ( F  |`  _V )  ~<_  ( F  |`  _V )
)
15 resss 5172 . . . 4  |-  ( F  |`  _V )  C_  F
16 ssdomg 7155 . . . 4  |-  ( F  e.  Fin  ->  (
( F  |`  _V )  C_  F  ->  ( F  |` 
_V )  ~<_  F ) )
1715, 16mpi 17 . . 3  |-  ( F  e.  Fin  ->  ( F  |`  _V )  ~<_  F )
18 domtr 7162 . . 3  |-  ( ( dom  ( F  |`  _V )  ~<_  ( F  |` 
_V )  /\  ( F  |`  _V )  ~<_  F )  ->  dom  ( F  |`  _V )  ~<_  F )
1914, 17, 18syl2anc 644 . 2  |-  ( F  e.  Fin  ->  dom  ( F  |`  _V )  ~<_  F )
201, 19syl5eqbrr 4248 1  |-  ( F  e.  Fin  ->  dom  F  ~<_  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   class class class wbr 4214    e. cmpt 4268   dom cdm 4880   ran crn 4881    |` cres 4882   Rel wrel 4885    Fn wfn 5451   -onto->wfo 5454   ` cfv 5456   1stc1st 6349    ~<_ cdom 7109   Fincfn 7111
This theorem is referenced by:  dmfi  7391  hashfun  11702
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1st 6351  df-2nd 6352  df-1o 6726  df-er 6907  df-en 7112  df-dom 7113  df-fin 7115
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