Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  inxprnres Structured version   Visualization version   GIF version

Theorem inxprnres 37891
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 5816 . 2 Rel (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))
2 relopabv 5823 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
3 eleq1w 2808 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
4 breq1 5152 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝑅𝑦𝑧𝑅𝑦))
53, 4anbi12d 630 . . . . 5 (𝑥 = 𝑧 → ((𝑥𝐴𝑥𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑦)))
6 breq2 5153 . . . . . 6 (𝑦 = 𝑤 → (𝑧𝑅𝑦𝑧𝑅𝑤))
76anbi2d 628 . . . . 5 (𝑦 = 𝑤 → ((𝑧𝐴𝑧𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑤)))
85, 7opelopabg 5540 . . . 4 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)} ↔ (𝑧𝐴𝑧𝑅𝑤)))
98el2v 3469 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)} ↔ (𝑧𝐴𝑧𝑅𝑤))
10 brinxprnres 37890 . . . 4 (𝑤 ∈ V → (𝑧(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝑤 ↔ (𝑧𝐴𝑧𝑅𝑤)))
1110elv 3467 . . 3 (𝑧(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝑤 ↔ (𝑧𝐴𝑧𝑅𝑤))
12 df-br 5150 . . 3 (𝑧(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑅 ∩ (𝐴 × ran (𝑅𝐴))))
139, 11, 123bitr2ri 299 . 2 (⟨𝑧, 𝑤⟩ ∈ (𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)})
141, 2, 13eqrelriiv 5792 1 (𝑅 ∩ (𝐴 × ran (𝑅𝐴))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wcel 2098  Vcvv 3461  cin 3943  cop 4636   class class class wbr 5149  {copab 5211   × cxp 5676  ran crn 5679  cres 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690
This theorem is referenced by:  dfres4  37892
  Copyright terms: Public domain W3C validator