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Theorem refsymrels2 35800
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 35822) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 35752, cf. the comment of dfrefrels2 35752. (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 35752 . . 3 RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2 dfsymrels2 35780 . . 3 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
31, 2ineq12i 4186 . 2 ( RefRels ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ 𝑟𝑟})
4 inrab 4274 . 2 ({𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ 𝑟𝑟}) = {𝑟 ∈ Rels ∣ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟𝑟𝑟)}
5 symrefref2 35798 . . . 4 (𝑟𝑟 → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ( I ↾ dom 𝑟) ⊆ 𝑟))
65pm5.32ri 578 . . 3 ((( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟𝑟𝑟) ↔ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟))
76rabbii 3473 . 2 {𝑟 ∈ Rels ∣ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟𝑟𝑟)} = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
83, 4, 73eqtri 2848 1 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1533  {crab 3142  cin 3934  wss 3935   I cid 5458   × cxp 5552  ccnv 5553  dom cdm 5554  ran crn 5555  cres 5556   Rels crels 35454   RefRels crefrels 35457   SymRels csymrels 35463
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-rels 35724  df-ssr 35737  df-refs 35749  df-refrels 35750  df-syms 35777  df-symrels 35778
This theorem is referenced by:  refsymrels3  35801  elrefsymrels2  35804  dfeqvrels2  35822  refrelsredund4  35866
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