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Theorem dfres4 37765
Description: Alternate definition of the restriction of a class. (Contributed by Peter Mazsa, 2-Jan-2019.)
Assertion
Ref Expression
dfres4 (𝑅𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))

Proof of Theorem dfres4
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfres2 6045 . 2 (𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
2 inxprnres 37764 . 2 (𝑅 ∩ (𝐴 × ran (𝑅𝐴))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
31, 2eqtr4i 2759 1 (𝑅𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1534  wcel 2099  cin 3946   class class class wbr 5148  {copab 5210   × cxp 5676  ran crn 5679  cres 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690
This theorem is referenced by:  xrnres4  37877  xrnresex  37878
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