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Theorem fi0 7691
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 4443 . . 3  |-  (/)  e.  _V
2 fival 7683 . . 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 4451 . . . 4  |-  -.  _V  e.  _V
5 id 22 . . . . . . 7  |-  ( y  =  |^| x  -> 
y  =  |^| x
)
6 inss1 3591 . . . . . . . . . . 11  |-  ( ~P (/)  i^i  Fin )  C_  ~P (/)
76sseli 3373 . . . . . . . . . 10  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  x  e.  ~P (/) )
8 elpwi 3890 . . . . . . . . . 10  |-  ( x  e.  ~P (/)  ->  x  C_  (/) )
9 ss0 3689 . . . . . . . . . 10  |-  ( x 
C_  (/)  ->  x  =  (/) )
107, 8, 93syl 20 . . . . . . . . 9  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  x  =  (/) )
1110inteqd 4154 . . . . . . . 8  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  |^| x  =  |^| (/) )
12 int0 4163 . . . . . . . 8  |-  |^| (/)  =  _V
1311, 12syl6eq 2491 . . . . . . 7  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  |^| x  =  _V )
145, 13sylan9eqr 2497 . . . . . 6  |-  ( ( x  e.  ( ~P (/)  i^i  Fin )  /\  y  =  |^| x )  ->  y  =  _V )
1514rexlimiva 2857 . . . . 5  |-  ( E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x  -> 
y  =  _V )
16 vex 2996 . . . . 5  |-  y  e. 
_V
1715, 16syl6eqelr 2532 . . . 4  |-  ( E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x  ->  _V  e.  _V )
184, 17mto 176 . . 3  |-  -.  E. x  e.  ( ~P (/) 
i^i  Fin ) y  = 
|^| x
1918abf 3692 . 2  |-  { y  |  E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x }  =  (/)
203, 19eqtri 2463 1  |-  ( fi
`  (/) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2737   _Vcvv 2993    i^i cin 3348    C_ wss 3349   (/)c0 3658   ~Pcpw 3881   |^|cint 4149   ` cfv 5439   Fincfn 7331   ficfi 7681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-int 4150  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-iota 5402  df-fun 5441  df-fv 5447  df-fi 7682
This theorem is referenced by:  fieq0  7692  firest  14392  restbas  18784
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