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Theorem fi0 7952
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 4528 . . 3  |-  (/)  e.  _V
2 fival 7944 . . 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 4534 . . . 4  |-  -.  _V  e.  _V
5 id 22 . . . . . . 7  |-  ( y  =  |^| x  -> 
y  =  |^| x
)
6 inss1 3643 . . . . . . . . . . 11  |-  ( ~P (/)  i^i  Fin )  C_  ~P (/)
76sseli 3414 . . . . . . . . . 10  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  x  e.  ~P (/) )
8 elpwi 3951 . . . . . . . . . 10  |-  ( x  e.  ~P (/)  ->  x  C_  (/) )
9 ss0 3768 . . . . . . . . . 10  |-  ( x 
C_  (/)  ->  x  =  (/) )
107, 8, 93syl 18 . . . . . . . . 9  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  x  =  (/) )
1110inteqd 4231 . . . . . . . 8  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  |^| x  =  |^| (/) )
12 int0 4240 . . . . . . . 8  |-  |^| (/)  =  _V
1311, 12syl6eq 2521 . . . . . . 7  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  |^| x  =  _V )
145, 13sylan9eqr 2527 . . . . . 6  |-  ( ( x  e.  ( ~P (/)  i^i  Fin )  /\  y  =  |^| x )  ->  y  =  _V )
1514rexlimiva 2868 . . . . 5  |-  ( E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x  -> 
y  =  _V )
16 vex 3034 . . . . 5  |-  y  e. 
_V
1715, 16syl6eqelr 2558 . . . 4  |-  ( E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x  ->  _V  e.  _V )
184, 17mto 181 . . 3  |-  -.  E. x  e.  ( ~P (/) 
i^i  Fin ) y  = 
|^| x
1918abf 3772 . 2  |-  { y  |  E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x }  =  (/)
203, 19eqtri 2493 1  |-  ( fi
`  (/) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452    e. wcel 1904   {cab 2457   E.wrex 2757   _Vcvv 3031    i^i cin 3389    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   |^|cint 4226   ` cfv 5589   Fincfn 7587   ficfi 7942
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-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-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-fi 7943
This theorem is referenced by:  fieq0  7953  firest  15409  restbas  20251
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