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Theorem iinfi 7357
Description: An indexed intersection of elements of  C is an element of the finite intersections of  C. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
iinfi  |-  ( ( C  e.  V  /\  ( A. x  e.  A  B  e.  C  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^|_ x  e.  A  B  e.  ( fi `  C
) )
Distinct variable groups:    x, A    x, C
Allowed substitution hints:    B( x)    V( x)

Proof of Theorem iinfi
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpr1 963 . . . 4  |-  ( ( C  e.  V  /\  ( A. x  e.  A  B  e.  C  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A. x  e.  A  B  e.  C )
2 dfiin2g 4066 . . . 4  |-  ( A. x  e.  A  B  e.  C  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B } )
31, 2syl 16 . . 3  |-  ( ( C  e.  V  /\  ( A. x  e.  A  B  e.  C  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B }
)
4 eqid 2387 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
54rnmpt 5056 . . . 4  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
65inteqi 3996 . . 3  |-  |^| ran  ( x  e.  A  |->  B )  =  |^| { y  |  E. x  e.  A  y  =  B }
73, 6syl6eqr 2437 . 2  |-  ( ( C  e.  V  /\  ( A. x  e.  A  B  e.  C  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )
84fmpt 5829 . . . 4  |-  ( A. x  e.  A  B  e.  C  <->  ( x  e.  A  |->  B ) : A --> C )
983anbi1i 1144 . . 3  |-  ( ( A. x  e.  A  B  e.  C  /\  A  =/=  (/)  /\  A  e. 
Fin )  <->  ( (
x  e.  A  |->  B ) : A --> C  /\  A  =/=  (/)  /\  A  e. 
Fin ) )
10 intrnfi 7356 . . 3  |-  ( ( C  e.  V  /\  ( ( x  e.  A  |->  B ) : A --> C  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| ran  ( x  e.  A  |->  B )  e.  ( fi `  C
) )
119, 10sylan2b 462 . 2  |-  ( ( C  e.  V  /\  ( A. x  e.  A  B  e.  C  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| ran  ( x  e.  A  |->  B )  e.  ( fi `  C
) )
127, 11eqeltrd 2461 1  |-  ( ( C  e.  V  /\  ( A. x  e.  A  B  e.  C  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^|_ x  e.  A  B  e.  ( fi `  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {cab 2373    =/= wne 2550   A.wral 2649   E.wrex 2650   (/)c0 3571   |^|cint 3992   |^|_ciin 4036    e. cmpt 4207   ran crn 4819   -->wf 5390   ` cfv 5394   Fincfn 7045   ficfi 7350
This theorem is referenced by:  firest  13587  iscmet3  19117
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-1o 6660  df-er 6841  df-en 7046  df-dom 7047  df-fin 7049  df-fi 7351
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