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Theorem dfiin3g 5086
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 4088 . 2  |-  ( A. x  e.  A  B  e.  C  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B } )
2 eqid 2408 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
32rnmpt 5079 . . 3  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
43inteqi 4018 . 2  |-  |^| ran  ( x  e.  A  |->  B )  =  |^| { y  |  E. x  e.  A  y  =  B }
51, 4syl6eqr 2458 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 1649    e. wcel 1721   {cab 2394   A.wral 2670   E.wrex 2671   |^|cint 4014   |^|_ciin 4058    e. cmpt 4230   ran crn 4842
This theorem is referenced by:  dfiin3  5088  riinint  5089  iinon  6565  cmpfi  17429
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-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-int 4015  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-cnv 4849  df-dm 4851  df-rn 4852
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