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Theorem inxprnres 36164
Description: Restriction of a class as a class of ordered pairs. (Contributed by Peter Mazsa, 2-Jan-2019.)
Assertion
Ref Expression
inxprnres (𝑅 ∩ (𝐴 × ran (𝑅𝐴))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦

Proof of Theorem inxprnres
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relinxp 5684 . 2 Rel (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))
2 relopabv 5691 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
3 eleq1w 2820 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
4 breq1 5056 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝑅𝑦𝑧𝑅𝑦))
53, 4anbi12d 634 . . . . 5 (𝑥 = 𝑧 → ((𝑥𝐴𝑥𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑦)))
6 breq2 5057 . . . . . 6 (𝑦 = 𝑤 → (𝑧𝑅𝑦𝑧𝑅𝑤))
76anbi2d 632 . . . . 5 (𝑦 = 𝑤 → ((𝑧𝐴𝑧𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑤)))
85, 7opelopabg 5419 . . . 4 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)} ↔ (𝑧𝐴𝑧𝑅𝑤)))
98el2v 3416 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)} ↔ (𝑧𝐴𝑧𝑅𝑤))
10 brinxprnres 36163 . . . 4 (𝑤 ∈ V → (𝑧(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝑤 ↔ (𝑧𝐴𝑧𝑅𝑤)))
1110elv 3414 . . 3 (𝑧(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝑤 ↔ (𝑧𝐴𝑧𝑅𝑤))
12 df-br 5054 . . 3 (𝑧(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑅 ∩ (𝐴 × ran (𝑅𝐴))))
139, 11, 123bitr2ri 303 . 2 (⟨𝑧, 𝑤⟩ ∈ (𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)})
141, 2, 13eqrelriiv 5660 1 (𝑅 ∩ (𝐴 × ran (𝑅𝐴))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  wcel 2110  Vcvv 3408  cin 3865  cop 4547   class class class wbr 5053  {copab 5115   × cxp 5549  ran crn 5552  cres 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559  df-dm 5561  df-rn 5562  df-res 5563
This theorem is referenced by:  dfres4  36165
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