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Theorem iinfi 7889
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 1002 . . . 4  |-  ( ( C  e.  V  /\  ( A. x  e.  A  B  e.  C  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A. x  e.  A  B  e.  C )
2 dfiin2g 4364 . . . 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 2467 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
54rnmpt 5254 . . . 4  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
65inteqi 4292 . . 3  |-  |^| ran  ( x  e.  A  |->  B )  =  |^| { y  |  E. x  e.  A  y  =  B }
73, 6syl6eqr 2526 . 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 6053 . . . 4  |-  ( A. x  e.  A  B  e.  C  <->  ( x  e.  A  |->  B ) : A --> C )
983anbi1i 1187 . . 3  |-  ( ( A. x  e.  A  B  e.  C  /\  A  =/=  (/)  /\  A  e. 
Fin )  <->  ( (
x  e.  A  |->  B ) : A --> C  /\  A  =/=  (/)  /\  A  e. 
Fin ) )
10 intrnfi 7888 . . 3  |-  ( ( C  e.  V  /\  ( ( x  e.  A  |->  B ) : A --> C  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| ran  ( x  e.  A  |->  B )  e.  ( fi `  C
) )
119, 10sylan2b 475 . 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 2555 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2817   E.wrex 2818   (/)c0 3790   |^|cint 4288   |^|_ciin 4332    |-> cmpt 4511   ran crn 5006   -->wf 5590   ` cfv 5594   Fincfn 7528   ficfi 7882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-1o 7142  df-er 7323  df-en 7529  df-dom 7530  df-fin 7532  df-fi 7883
This theorem is referenced by:  firest  14705  iscmet3  21600
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