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Theorem refsymrels2 38547
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 dfeqvrels2 38570) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 38495, cf. the comment of dfrefrels2 38495. (Contributed by Peter Mazsa, 20-Jul-2019.)
Assertion
Ref Expression
refsymrels2 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}

Proof of Theorem refsymrels2
StepHypRef Expression
1 dfrefrels2 38495 . . 3 RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2 dfsymrels2 38527 . . 3 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
31, 2ineq12i 4226 . 2 ( RefRels ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ 𝑟𝑟})
4 inrab 4322 . 2 ({𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ 𝑟𝑟}) = {𝑟 ∈ Rels ∣ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟𝑟𝑟)}
5 symrefref2 38545 . . . 4 (𝑟𝑟 → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ( I ↾ dom 𝑟) ⊆ 𝑟))
65pm5.32ri 575 . . 3 ((( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟𝑟𝑟) ↔ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟))
76rabbii 3439 . 2 {𝑟 ∈ Rels ∣ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟𝑟𝑟)} = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
83, 4, 73eqtri 2767 1 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  {crab 3433  cin 3962  wss 3963   I cid 5582   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691   Rels crels 38164   RefRels crefrels 38167   SymRels csymrels 38173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-rels 38467  df-ssr 38480  df-refs 38492  df-refrels 38493  df-syms 38524  df-symrels 38525
This theorem is referenced by:  refsymrels3  38548  elrefsymrels2  38551  dfeqvrels2  38570  refrelsredund4  38614
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