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Theorem inxprnres 34384
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 relxp 5283 . . 3 Rel (𝐴 × ran (𝑅𝐴))
2 relin2 5393 . . 3 (Rel (𝐴 × ran (𝑅𝐴)) → Rel (𝑅 ∩ (𝐴 × ran (𝑅𝐴))))
31, 2ax-mp 5 . 2 Rel (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))
4 relopab 5403 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
5 eleq1w 2822 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
6 breq1 4807 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝑅𝑦𝑧𝑅𝑦))
75, 6anbi12d 749 . . . . 5 (𝑥 = 𝑧 → ((𝑥𝐴𝑥𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑦)))
8 breq2 4808 . . . . . 6 (𝑦 = 𝑤 → (𝑧𝑅𝑦𝑧𝑅𝑤))
98anbi2d 742 . . . . 5 (𝑦 = 𝑤 → ((𝑧𝐴𝑧𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑤)))
107, 9opelopabg 5143 . . . 4 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)} ↔ (𝑧𝐴𝑧𝑅𝑤)))
1110el2v 34310 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)} ↔ (𝑧𝐴𝑧𝑅𝑤))
12 brinxprnres 34383 . . . 4 (𝑤 ∈ V → (𝑧(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝑤 ↔ (𝑧𝐴𝑧𝑅𝑤)))
1312elv 34309 . . 3 (𝑧(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝑤 ↔ (𝑧𝐴𝑧𝑅𝑤))
14 df-br 4805 . . 3 (𝑧(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑅 ∩ (𝐴 × ran (𝑅𝐴))))
1511, 13, 143bitr2ri 289 . 2 (⟨𝑧, 𝑤⟩ ∈ (𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)})
163, 4, 15eqrelriiv 5371 1 (𝑅 ∩ (𝐴 × ran (𝑅𝐴))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cin 3714  cop 4327   class class class wbr 4804  {copab 4864   × cxp 5264  ran crn 5267  cres 5268  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278
This theorem is referenced by:  dfres4  34385
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