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Theorem fi0 7418
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 4332 . . 3  |-  (/)  e.  _V
2 fival 7410 . . 3  |-  ( (/)  e.  _V  ->  ( fi `  (/) )  =  {
y  |  E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x } )
31, 2ax-mp 8 . 2  |-  ( fi
`  (/) )  =  {
y  |  E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x }
4 vprc 4334 . . . 4  |-  -.  _V  e.  _V
5 id 20 . . . . . . 7  |-  ( y  =  |^| x  -> 
y  =  |^| x
)
6 inss1 3554 . . . . . . . . . . 11  |-  ( ~P (/)  i^i  Fin )  C_  ~P (/)
76sseli 3337 . . . . . . . . . 10  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  x  e.  ~P (/) )
8 elpwi 3800 . . . . . . . . . 10  |-  ( x  e.  ~P (/)  ->  x  C_  (/) )
9 ss0 3651 . . . . . . . . . 10  |-  ( x 
C_  (/)  ->  x  =  (/) )
107, 8, 93syl 19 . . . . . . . . 9  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  x  =  (/) )
1110inteqd 4048 . . . . . . . 8  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  |^| x  =  |^| (/) )
12 int0 4057 . . . . . . . 8  |-  |^| (/)  =  _V
1311, 12syl6eq 2484 . . . . . . 7  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  |^| x  =  _V )
145, 13sylan9eqr 2490 . . . . . 6  |-  ( ( x  e.  ( ~P (/)  i^i  Fin )  /\  y  =  |^| x )  ->  y  =  _V )
1514rexlimiva 2818 . . . . 5  |-  ( E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x  -> 
y  =  _V )
16 vex 2952 . . . . 5  |-  y  e. 
_V
1715, 16syl6eqelr 2525 . . . 4  |-  ( E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x  ->  _V  e.  _V )
184, 17mto 169 . . 3  |-  -.  E. x  e.  ( ~P (/) 
i^i  Fin ) y  = 
|^| x
1918abf 3654 . 2  |-  { y  |  E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x }  =  (/)
203, 19eqtri 2456 1  |-  ( fi
`  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   {cab 2422   E.wrex 2699   _Vcvv 2949    i^i cin 3312    C_ wss 3313   (/)c0 3621   ~Pcpw 3792   |^|cint 4043   ` cfv 5447   Fincfn 7102   ficfi 7408
This theorem is referenced by:  fieq0  7419  firest  13653  restbas  17215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-int 4044  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-iota 5411  df-fun 5449  df-fv 5455  df-fi 7409
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