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Theorem inxprnres 35860
 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 5655 . 2 Rel (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))
2 relopab 5664 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
3 eleq1w 2872 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
4 breq1 5037 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝑅𝑦𝑧𝑅𝑦))
53, 4anbi12d 633 . . . . 5 (𝑥 = 𝑧 → ((𝑥𝐴𝑥𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑦)))
6 breq2 5038 . . . . . 6 (𝑦 = 𝑤 → (𝑧𝑅𝑦𝑧𝑅𝑤))
76anbi2d 631 . . . . 5 (𝑦 = 𝑤 → ((𝑧𝐴𝑧𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑤)))
85, 7opelopabg 5394 . . . 4 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)} ↔ (𝑧𝐴𝑧𝑅𝑤)))
98el2v 3449 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)} ↔ (𝑧𝐴𝑧𝑅𝑤))
10 brinxprnres 35859 . . . 4 (𝑤 ∈ V → (𝑧(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝑤 ↔ (𝑧𝐴𝑧𝑅𝑤)))
1110elv 3447 . . 3 (𝑧(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝑤 ↔ (𝑧𝐴𝑧𝑅𝑤))
12 df-br 5035 . . 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 5631 1 (𝑅 ∩ (𝐴 × ran (𝑅𝐴))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3442   ∩ cin 3882  ⟨cop 4534   class class class wbr 5034  {copab 5096   × cxp 5521  ran crn 5524   ↾ cres 5525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535 This theorem is referenced by:  dfres4  35861
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