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Theorem refsymrels2 34804
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 34825) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 34756, cf. the comment of dfrefrels2 34756. (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 34756 . . 3 RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2 dfsymrels2 34784 . . 3 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
31, 2ineq12i 4011 . 2 ( RefRels ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ 𝑟𝑟})
4 inrab 4100 . 2 ({𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ 𝑟𝑟}) = {𝑟 ∈ Rels ∣ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟𝑟𝑟)}
5 symrefref2 34802 . . . 4 (𝑟𝑟 → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ( I ↾ dom 𝑟) ⊆ 𝑟))
65pm5.32ri 572 . . 3 ((( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟𝑟𝑟) ↔ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟))
76rabbii 3370 . 2 {𝑟 ∈ Rels ∣ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟𝑟𝑟)} = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
83, 4, 73eqtri 2826 1 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
Colors of variables: wff setvar class
Syntax hints:  wa 385   = wceq 1653  {crab 3094  cin 3769  wss 3770   I cid 5220   × cxp 5311  ccnv 5312  dom cdm 5313  ran crn 5314  cres 5315   Rels crels 34470   RefRels crefrels 34473   SymRels csymrels 34479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-br 4845  df-opab 4907  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-dm 5323  df-rn 5324  df-res 5325  df-rels 34728  df-ssr 34741  df-refs 34753  df-refrels 34754  df-syms 34781  df-symrels 34782
This theorem is referenced by:  refsymrels3  34805  elrefsymrels2  34808  dfeqvrels2  34825  refrelsred4  34866
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