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Theorem dfiin3g 5080
Description: Alternate definition of indexed intersection when  B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
dfiin3g  |-  ( A. x  e.  A  B  e.  C  ->  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )

Proof of Theorem dfiin3g
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfiin2g 4191 . 2  |-  ( A. x  e.  A  B  e.  C  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B } )
2 eqid 2433 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
32rnmpt 5072 . . 3  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
43inteqi 4120 . 2  |-  |^| ran  ( x  e.  A  |->  B )  =  |^| { y  |  E. x  e.  A  y  =  B }
51, 4syl6eqr 2483 1  |-  ( A. x  e.  A  B  e.  C  ->  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1362    e. wcel 1755   {cab 2419   A.wral 2705   E.wrex 2706   |^|cint 4116   |^|_ciin 4160    e. cmpt 4338   ran crn 4828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-int 4117  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-cnv 4835  df-dm 4837  df-rn 4838
This theorem is referenced by:  dfiin3  5082  riinint  5083  iinon  6787  cmpfi  18852
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