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Theorem dfiin3g 5169
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 4276 . 2  |-  ( A. x  e.  A  B  e.  C  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B } )
2 eqid 2382 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
32rnmpt 5161 . . 3  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
43inteqi 4203 . 2  |-  |^| ran  ( x  e.  A  |->  B )  =  |^| { y  |  E. x  e.  A  y  =  B }
51, 4syl6eqr 2441 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 1399    e. wcel 1826   {cab 2367   A.wral 2732   E.wrex 2733   |^|cint 4199   |^|_ciin 4244    |-> cmpt 4425   ran crn 4914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-int 4200  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-cnv 4921  df-dm 4923  df-rn 4924
This theorem is referenced by:  dfiin3  5171  riinint  5172  iinon  6929  cmpfi  19994
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