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Theorem inelfi 7664
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 4536 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  { A ,  B }  e.  ~P X
)
213adant1 1001 . . . 4  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  { A ,  B }  e.  ~P X
)
3 prfi 7582 . . . . 5  |-  { A ,  B }  e.  Fin
43a1i 11 . . . 4  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  { A ,  B }  e.  Fin )
52, 4elind 3537 . . 3  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  { A ,  B }  e.  ( ~P X  i^i  Fin ) )
6 intprg 4159 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
763adant1 1001 . . . 4  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
87eqcomd 2446 . . 3  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A  i^i  B
)  =  |^| { A ,  B } )
9 inteq 4128 . . . . 5  |-  ( p  =  { A ,  B }  ->  |^| p  =  |^| { A ,  B } )
109eqeq2d 2452 . . . 4  |-  ( p  =  { A ,  B }  ->  ( ( A  i^i  B )  =  |^| p  <->  ( A  i^i  B )  =  |^| { A ,  B }
) )
1110rspcev 3070 . . 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 656 . 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 4432 . . . 4  |-  ( A  e.  X  ->  ( A  i^i  B )  e. 
_V )
14133ad2ant2 1005 . . 3  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A  i^i  B
)  e.  _V )
15 simp1 983 . . 3  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  X  e.  V )
16 elfi 7659 . . 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 656 . 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 232 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 184    /\ w3a 960    = wceq 1364    e. wcel 1761   E.wrex 2714   _Vcvv 2970    i^i cin 3324   ~Pcpw 3857   {cpr 3876   |^|cint 4125   ` cfv 5415   Fincfn 7306   ficfi 7656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-fin 7310  df-fi 7657
This theorem is referenced by:  neiptoptop  18694
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