Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  refsymrels3 Structured version   Visualization version   GIF version

Theorem refsymrels3 37528
Description: Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels3 37551) can use the 𝑥 ∈ dom 𝑟𝑥𝑟𝑥 version for their reflexive part, not just the 𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦) version of dfrefrels3 37476, cf. the comment of dfrefrel3 37478. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
refsymrels3 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥))}
Distinct variable group:   𝑥,𝑟,𝑦

Proof of Theorem refsymrels3
StepHypRef Expression
1 refsymrels2 37527 . 2 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
2 idrefALT 6112 . . 3 (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥)
3 cnvsym 6113 . . 3 (𝑟𝑟 ↔ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥))
42, 3anbi12i 627 . 2 ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ↔ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥)))
51, 4rabbieq 37210 1 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥))}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539   = wceq 1541  wral 3061  {crab 3432  cin 3947  wss 3948   class class class wbr 5148   I cid 5573  ccnv 5675  dom cdm 5676  cres 5678   Rels crels 37137   RefRels crefrels 37140   SymRels csymrels 37146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-rels 37447  df-ssr 37460  df-refs 37472  df-refrels 37473  df-syms 37504  df-symrels 37505
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator