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

Theorem rnxrnidres 37575
Description: Range of a range Cartesian product with a restriction of the identity relation. (Contributed by Peter Mazsa, 6-Dec-2021.)
Assertion
Ref Expression
rnxrnidres ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 = 𝑦𝑢𝑅𝑥)}
Distinct variable groups:   𝑢,𝐴,𝑥,𝑦   𝑢,𝑅,𝑥,𝑦

Proof of Theorem rnxrnidres
StepHypRef Expression
1 rnxrnres 37573 . 2 ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢 I 𝑦)}
2 ideqg 5851 . . . . . 6 (𝑦 ∈ V → (𝑢 I 𝑦𝑢 = 𝑦))
32elv 3479 . . . . 5 (𝑢 I 𝑦𝑢 = 𝑦)
43anbi1ci 625 . . . 4 ((𝑢𝑅𝑥𝑢 I 𝑦) ↔ (𝑢 = 𝑦𝑢𝑅𝑥))
54rexbii 3093 . . 3 (∃𝑢𝐴 (𝑢𝑅𝑥𝑢 I 𝑦) ↔ ∃𝑢𝐴 (𝑢 = 𝑦𝑢𝑅𝑥))
65opabbii 5215 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢 I 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 = 𝑦𝑢𝑅𝑥)}
71, 6eqtri 2759 1 ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 = 𝑦𝑢𝑅𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1540  wrex 3069  Vcvv 3473   class class class wbr 5148  {copab 5210   I cid 5573  ran crn 5677  cres 5678  cxrn 37346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7979  df-2nd 7980  df-ec 8709  df-xrn 37545
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator