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Theorem iinfi 7380
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 4084 . . . 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 2404 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
54rnmpt 5075 . . . 4  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
65inteqi 4014 . . 3  |-  |^| ran  ( x  e.  A  |->  B )  =  |^| { y  |  E. x  e.  A  y  =  B }
73, 6syl6eqr 2454 . 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 5849 . . . 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 7379 . . 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 2478 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 1721   {cab 2390    =/= wne 2567   A.wral 2666   E.wrex 2667   (/)c0 3588   |^|cint 4010   |^|_ciin 4054    e. cmpt 4226   ran crn 4838   -->wf 5409   ` cfv 5413   Fincfn 7068   ficfi 7373
This theorem is referenced by:  firest  13615  iscmet3  19199
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-fin 7072  df-fi 7374
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