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Theorem refsymrels2 35681
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 35703) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 35633, cf. the comment of dfrefrels2 35633. (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 35633 . . 3 RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2 dfsymrels2 35661 . . 3 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
31, 2ineq12i 4184 . 2 ( RefRels ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ 𝑟𝑟})
4 inrab 4272 . 2 ({𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ 𝑟𝑟}) = {𝑟 ∈ Rels ∣ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟𝑟𝑟)}
5 symrefref2 35679 . . . 4 (𝑟𝑟 → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ( I ↾ dom 𝑟) ⊆ 𝑟))
65pm5.32ri 576 . . 3 ((( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟𝑟𝑟) ↔ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟))
76rabbii 3471 . 2 {𝑟 ∈ Rels ∣ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟𝑟𝑟)} = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
83, 4, 73eqtri 2845 1 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  {crab 3139  cin 3932  wss 3933   I cid 5452   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  cres 5550   Rels crels 35336   RefRels crefrels 35339   SymRels csymrels 35345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-rels 35605  df-ssr 35618  df-refs 35630  df-refrels 35631  df-syms 35658  df-symrels 35659
This theorem is referenced by:  refsymrels3  35682  elrefsymrels2  35685  dfeqvrels2  35703  refrelsredund4  35747
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