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Theorem dfres4 38274
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 6060 . 2 (𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
2 inxprnres 38273 . 2 (𝑅 ∩ (𝐴 × ran (𝑅𝐴))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
31, 2eqtr4i 2765 1 (𝑅𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1536  wcel 2105  cin 3961   class class class wbr 5147  {copab 5209   × cxp 5686  ran crn 5689  cres 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700
This theorem is referenced by:  xrnres4  38386  xrnresex  38387
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