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 34401
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 34399 . 2 ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢 I 𝑦)}
2 ideqg 5381 . . . . . 6 (𝑦 ∈ V → (𝑢 I 𝑦𝑢 = 𝑦))
32elv 34228 . . . . 5 (𝑢 I 𝑦𝑢 = 𝑦)
43anbi1ci 34241 . . . 4 ((𝑢𝑅𝑥𝑢 I 𝑦) ↔ (𝑢 = 𝑦𝑢𝑅𝑥))
54rexbii 3143 . . 3 (∃𝑢𝐴 (𝑢𝑅𝑥𝑢 I 𝑦) ↔ ∃𝑢𝐴 (𝑢 = 𝑦𝑢𝑅𝑥))
65opabbii 4825 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢 I 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 = 𝑦𝑢𝑅𝑥)}
71, 6eqtri 2746 1 ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 = 𝑦𝑢𝑅𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1596  wrex 3015  Vcvv 3304   class class class wbr 4760  {copab 4820   I cid 5127  ran crn 5219  cres 5220  cxrn 34214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-fo 6007  df-fv 6009  df-1st 7285  df-2nd 7286  df-ec 7864  df-xrn 34375
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator