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Theorem inxprnres 34990
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 5537 . 2 Rel (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))
2 relopab 5546 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
3 eleq1w 2849 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
4 breq1 4932 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝑅𝑦𝑧𝑅𝑦))
53, 4anbi12d 621 . . . . 5 (𝑥 = 𝑧 → ((𝑥𝐴𝑥𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑦)))
6 breq2 4933 . . . . . 6 (𝑦 = 𝑤 → (𝑧𝑅𝑦𝑧𝑅𝑤))
76anbi2d 619 . . . . 5 (𝑦 = 𝑤 → ((𝑧𝐴𝑧𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑤)))
85, 7opelopabg 5279 . . . 4 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)} ↔ (𝑧𝐴𝑧𝑅𝑤)))
98el2v 3423 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)} ↔ (𝑧𝐴𝑧𝑅𝑤))
10 brinxprnres 34989 . . . 4 (𝑤 ∈ V → (𝑧(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝑤 ↔ (𝑧𝐴𝑧𝑅𝑤)))
1110elv 3421 . . 3 (𝑧(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝑤 ↔ (𝑧𝐴𝑧𝑅𝑤))
12 df-br 4930 . . 3 (𝑧(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑅 ∩ (𝐴 × ran (𝑅𝐴))))
139, 11, 123bitr2ri 292 . 2 (⟨𝑧, 𝑤⟩ ∈ (𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)})
141, 2, 13eqrelriiv 5513 1 (𝑅 ∩ (𝐴 × ran (𝑅𝐴))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387   = wceq 1507  wcel 2050  Vcvv 3416  cin 3829  cop 4447   class class class wbr 4929  {copab 4991   × cxp 5405  ran crn 5408  cres 5409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992  df-xp 5413  df-rel 5414  df-cnv 5415  df-dm 5417  df-rn 5418  df-res 5419
This theorem is referenced by:  dfres4  34991
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