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Theorem iinfi 7171
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 961 . . . 4  |-  ( ( C  e.  V  /\  ( A. x  e.  A  B  e.  C  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A. x  e.  A  B  e.  C )
2 dfiin2g 3936 . . . 4  |-  ( A. x  e.  A  B  e.  C  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B } )
31, 2syl 15 . . 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 2283 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
54rnmpt 4925 . . . 4  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
65inteqi 3866 . . 3  |-  |^| ran  ( x  e.  A  |->  B )  =  |^| { y  |  E. x  e.  A  y  =  B }
73, 6syl6eqr 2333 . 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 5681 . . . 4  |-  ( A. x  e.  A  B  e.  C  <->  ( x  e.  A  |->  B ) : A --> C )
983anbi1i 1142 . . 3  |-  ( ( A. x  e.  A  B  e.  C  /\  A  =/=  (/)  /\  A  e. 
Fin )  <->  ( (
x  e.  A  |->  B ) : A --> C  /\  A  =/=  (/)  /\  A  e. 
Fin ) )
10 intrnfi 7170 . . 3  |-  ( ( C  e.  V  /\  ( ( x  e.  A  |->  B ) : A --> C  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| ran  ( x  e.  A  |->  B )  e.  ( fi `  C
) )
119, 10sylan2b 461 . 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 2357 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   (/)c0 3455   |^|cint 3862   |^|_ciin 3906    e. cmpt 4077   ran crn 4690   -->wf 5251   ` cfv 5255   Fincfn 6863   ficfi 7164
This theorem is referenced by:  firest  13337  iscmet3  18719
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-fin 6867  df-fi 7165
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