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Theorem fidomdm 7835
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 5281 . 2  |-  dom  ( F  |`  _V )  =  dom  F
2 finresfin 7779 . . . 4  |-  ( F  e.  Fin  ->  ( F  |`  _V )  e. 
Fin )
3 fvex 5858 . . . . . . 7  |-  ( 1st `  x )  e.  _V
4 eqid 2402 . . . . . . 7  |-  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) )  =  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x
) )
53, 4fnmpti 5691 . . . . . 6  |-  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) )  Fn  ( F  |`  _V )
6 dffn4 5783 . . . . . 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 208 . . . . 5  |-  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) ) : ( F  |`  _V ) -onto-> ran  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x
) )
8 relres 5120 . . . . . 6  |-  Rel  ( F  |`  _V )
9 reldm 6834 . . . . . 6  |-  ( Rel  ( F  |`  _V )  ->  dom  ( F  |`  _V )  =  ran  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) ) )
10 foeq3 5775 . . . . . 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 209 . . . 4  |-  ( x  e.  ( F  |`  _V )  |->  ( 1st `  x ) ) : ( F  |`  _V ) -onto-> dom  ( F  |`  _V )
13 fodomfi 7832 . . . 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 660 . . 3  |-  ( F  e.  Fin  ->  dom  ( F  |`  _V )  ~<_  ( F  |`  _V )
)
15 resss 5116 . . . 4  |-  ( F  |`  _V )  C_  F
16 ssdomg 7598 . . . 4  |-  ( F  e.  Fin  ->  (
( F  |`  _V )  C_  F  ->  ( F  |` 
_V )  ~<_  F ) )
1715, 16mpi 18 . . 3  |-  ( F  e.  Fin  ->  ( F  |`  _V )  ~<_  F )
18 domtr 7605 . . 3  |-  ( ( dom  ( F  |`  _V )  ~<_  ( F  |` 
_V )  /\  ( F  |`  _V )  ~<_  F )  ->  dom  ( F  |`  _V )  ~<_  F )
1914, 17, 18syl2anc 659 . 2  |-  ( F  e.  Fin  ->  dom  ( F  |`  _V )  ~<_  F )
201, 19syl5eqbrr 4428 1  |-  ( F  e.  Fin  ->  dom  F  ~<_  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1405    e. wcel 1842   _Vcvv 3058    C_ wss 3413   class class class wbr 4394    |-> cmpt 4452   dom cdm 4822   ran crn 4823    |` cres 4824   Rel wrel 4827    Fn wfn 5563   -onto->wfo 5566   ` cfv 5568   1stc1st 6781    ~<_ cdom 7551   Fincfn 7553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-om 6683  df-1st 6783  df-2nd 6784  df-1o 7166  df-er 7347  df-en 7554  df-dom 7555  df-fin 7557
This theorem is referenced by:  dmfi  7836  hashfun  12542
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