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Theorem dfres4 38294
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 6059 . 2 (𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
2 inxprnres 38293 . 2 (𝑅 ∩ (𝐴 × ran (𝑅𝐴))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
31, 2eqtr4i 2768 1 (𝑅𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108  cin 3950   class class class wbr 5143  {copab 5205   × cxp 5683  ran crn 5686  cres 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697
This theorem is referenced by:  xrnres4  38406  xrnresex  38407
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