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Theorem dfres4 37991
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 6050 . 2 (𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
2 inxprnres 37990 . 2 (𝑅 ∩ (𝐴 × ran (𝑅𝐴))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
31, 2eqtr4i 2757 1 (𝑅𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1534  wcel 2099  cin 3946   class class class wbr 5153  {copab 5215   × cxp 5680  ran crn 5683  cres 5684
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 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694
This theorem is referenced by:  xrnres4  38103  xrnresex  38104
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