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Theorem dfiin3g 5126
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 4126 . 2  |-  ( A. x  e.  A  B  e.  C  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B } )
2 eqid 2438 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
32rnmpt 5119 . . 3  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
43inteqi 4056 . 2  |-  |^| ran  ( x  e.  A  |->  B )  =  |^| { y  |  E. x  e.  A  y  =  B }
51, 4syl6eqr 2488 1  |-  ( A. x  e.  A  B  e.  C  ->  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708   |^|cint 4052   |^|_ciin 4096    e. cmpt 4269   ran crn 4882
This theorem is referenced by:  dfiin3  5128  riinint  5129  iinon  6605  cmpfi  17476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-int 4053  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-cnv 4889  df-dm 4891  df-rn 4892
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