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Theorem fi0 7189
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 4166 . . 3  |-  (/)  e.  _V
2 fival 7182 . . 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 4168 . . . 4  |-  -.  _V  e.  _V
5 id 19 . . . . . . 7  |-  ( y  =  |^| x  -> 
y  =  |^| x
)
6 inss1 3402 . . . . . . . . . . 11  |-  ( ~P (/)  i^i  Fin )  C_  ~P (/)
76sseli 3189 . . . . . . . . . 10  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  x  e.  ~P (/) )
8 elpwi 3646 . . . . . . . . . 10  |-  ( x  e.  ~P (/)  ->  x  C_  (/) )
9 ss0 3498 . . . . . . . . . 10  |-  ( x 
C_  (/)  ->  x  =  (/) )
107, 8, 93syl 18 . . . . . . . . 9  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  x  =  (/) )
1110inteqd 3883 . . . . . . . 8  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  |^| x  =  |^| (/) )
12 int0 3892 . . . . . . . 8  |-  |^| (/)  =  _V
1311, 12syl6eq 2344 . . . . . . 7  |-  ( x  e.  ( ~P (/)  i^i  Fin )  ->  |^| x  =  _V )
145, 13sylan9eqr 2350 . . . . . 6  |-  ( ( x  e.  ( ~P (/)  i^i  Fin )  /\  y  =  |^| x )  ->  y  =  _V )
1514rexlimiva 2675 . . . . 5  |-  ( E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x  -> 
y  =  _V )
16 vex 2804 . . . . 5  |-  y  e. 
_V
1715, 16syl6eqelr 2385 . . . 4  |-  ( E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x  ->  _V  e.  _V )
184, 17mto 167 . . 3  |-  -.  E. x  e.  ( ~P (/) 
i^i  Fin ) y  = 
|^| x
1918abf 3501 . 2  |-  { y  |  E. x  e.  ( ~P (/)  i^i  Fin ) y  =  |^| x }  =  (/)
203, 19eqtri 2316 1  |-  ( fi
`  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   |^|cint 3878   ` cfv 5271   Fincfn 6879   ficfi 7180
This theorem is referenced by:  fieq0  7190  firest  13353  restbas  16905
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-fi 7181
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