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Theorem inelfi 7950
Description: The intersection of two sets is a finite intersection. (Contributed by Thierry Arnoux, 6-Jan-2017.)
Assertion
Ref Expression
inelfi  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A  i^i  B
)  e.  ( fi
`  X ) )

Proof of Theorem inelfi
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 prelpwi 4647 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  { A ,  B }  e.  ~P X
)
213adant1 1048 . . . 4  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  { A ,  B }  e.  ~P X
)
3 prfi 7864 . . . . 5  |-  { A ,  B }  e.  Fin
43a1i 11 . . . 4  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  { A ,  B }  e.  Fin )
52, 4elind 3609 . . 3  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  { A ,  B }  e.  ( ~P X  i^i  Fin ) )
6 intprg 4260 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
763adant1 1048 . . . 4  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
87eqcomd 2477 . . 3  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A  i^i  B
)  =  |^| { A ,  B } )
9 inteq 4229 . . . . 5  |-  ( p  =  { A ,  B }  ->  |^| p  =  |^| { A ,  B } )
109eqeq2d 2481 . . . 4  |-  ( p  =  { A ,  B }  ->  ( ( A  i^i  B )  =  |^| p  <->  ( A  i^i  B )  =  |^| { A ,  B }
) )
1110rspcev 3136 . . 3  |-  ( ( { A ,  B }  e.  ( ~P X  i^i  Fin )  /\  ( A  i^i  B )  =  |^| { A ,  B } )  ->  E. p  e.  ( ~P X  i^i  Fin )
( A  i^i  B
)  =  |^| p
)
125, 8, 11syl2anc 673 . 2  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  E. p  e.  ( ~P X  i^i  Fin ) ( A  i^i  B )  =  |^| p
)
13 inex1g 4539 . . . 4  |-  ( A  e.  X  ->  ( A  i^i  B )  e. 
_V )
14133ad2ant2 1052 . . 3  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A  i^i  B
)  e.  _V )
15 simp1 1030 . . 3  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  X  e.  V )
16 elfi 7945 . . 3  |-  ( ( ( A  i^i  B
)  e.  _V  /\  X  e.  V )  ->  ( ( A  i^i  B )  e.  ( fi
`  X )  <->  E. p  e.  ( ~P X  i^i  Fin ) ( A  i^i  B )  =  |^| p
) )
1714, 15, 16syl2anc 673 . 2  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A  i^i  B )  e.  ( fi
`  X )  <->  E. p  e.  ( ~P X  i^i  Fin ) ( A  i^i  B )  =  |^| p
) )
1812, 17mpbird 240 1  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A  i^i  B
)  e.  ( fi
`  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ w3a 1007    = wceq 1452    e. wcel 1904   E.wrex 2757   _Vcvv 3031    i^i cin 3389   ~Pcpw 3942   {cpr 3961   |^|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-3or 1008  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-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-fin 7591  df-fi 7943
This theorem is referenced by:  neiptoptop  20224  sigapildsyslem  29057
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