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Theorem inelfi 7683
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 4554 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  { A ,  B }  e.  ~P X
)
213adant1 1006 . . . 4  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  { A ,  B }  e.  ~P X
)
3 prfi 7601 . . . . 5  |-  { A ,  B }  e.  Fin
43a1i 11 . . . 4  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  { A ,  B }  e.  Fin )
52, 4elind 3555 . . 3  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  { A ,  B }  e.  ( ~P X  i^i  Fin ) )
6 intprg 4177 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
763adant1 1006 . . . 4  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
87eqcomd 2448 . . 3  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A  i^i  B
)  =  |^| { A ,  B } )
9 inteq 4146 . . . . 5  |-  ( p  =  { A ,  B }  ->  |^| p  =  |^| { A ,  B } )
109eqeq2d 2454 . . . 4  |-  ( p  =  { A ,  B }  ->  ( ( A  i^i  B )  =  |^| p  <->  ( A  i^i  B )  =  |^| { A ,  B }
) )
1110rspcev 3088 . . 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 661 . 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 4450 . . . 4  |-  ( A  e.  X  ->  ( A  i^i  B )  e. 
_V )
14133ad2ant2 1010 . . 3  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A  i^i  B
)  e.  _V )
15 simp1 988 . . 3  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  X  e.  V )
16 elfi 7678 . . 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 661 . 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 965    = wceq 1369    e. wcel 1756   E.wrex 2731   _Vcvv 2987    i^i cin 3342   ~Pcpw 3875   {cpr 3894   |^|cint 4143   ` cfv 5433   Fincfn 7325   ficfi 7675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-en 7326  df-fin 7329  df-fi 7676
This theorem is referenced by:  neiptoptop  18750
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