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Theorem fi0 7940
Description: The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
fi0  |-  ( fi
`  (/) )  =  (/)

Proof of Theorem fi0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4557 . . 3  |-  (/)  e.  _V
2 fival 7932 . . 3  |-  ( (/)  e.  _V  ->  ( fi `  (/) )  =  {
y  |  E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x } )
31, 2ax-mp 5 . 2  |-  ( fi
`  (/) )  =  {
y  |  E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x }
4 vprc 4563 . . . 4  |-  -.  _V  e.  _V
5 id 23 . . . . . . 7  |-  ( y  =  |^| x  -> 
y  =  |^| x
)
6 inss1 3688 . . . . . . . . . . 11  |-  ( ~P (/)  i^i  Fin )  C_  ~P (/)
76sseli 3466 . . . . . . . . . 10  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  x  e.  ~P (/) )
8 elpwi 3994 . . . . . . . . . 10  |-  ( x  e.  ~P (/)  ->  x  C_  (/) )
9 ss0 3799 . . . . . . . . . 10  |-  ( x 
C_  (/)  ->  x  =  (/) )
107, 8, 93syl 18 . . . . . . . . 9  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  x  =  (/) )
1110inteqd 4263 . . . . . . . 8  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  |^| x  =  |^| (/) )
12 int0 4272 . . . . . . . 8  |-  |^| (/)  =  _V
1311, 12syl6eq 2486 . . . . . . 7  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  |^| x  =  _V )
145, 13sylan9eqr 2492 . . . . . 6  |-  ( ( x  e.  ( ~P (/)  i^i  Fin )  /\  y  =  |^| x )  ->  y  =  _V )
1514rexlimiva 2920 . . . . 5  |-  ( E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x  -> 
y  =  _V )
16 vex 3090 . . . . 5  |-  y  e. 
_V
1715, 16syl6eqelr 2526 . . . 4  |-  ( E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x  ->  _V  e.  _V )
184, 17mto 179 . . 3  |-  -.  E. x  e.  ( ~P (/) 
i^i  Fin ) y  = 
|^| x
1918abf 3802 . 2  |-  { y  |  E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x }  =  (/)
203, 19eqtri 2458 1  |-  ( fi
`  (/) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1870   {cab 2414   E.wrex 2783   _Vcvv 3087    i^i cin 3441    C_ wss 3442   (/)c0 3767   ~Pcpw 3985   |^|cint 4258   ` cfv 5601   Fincfn 7577   ficfi 7930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-fi 7931
This theorem is referenced by:  fieq0  7941  firest  15290  restbas  20105
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