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Theorem elrefsymrels2 38551
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, 22-Jul-2019.)
Assertion
Ref Expression
elrefsymrels2 (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ 𝑅 ∈ Rels ))

Proof of Theorem elrefsymrels2
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 refsymrels2 38547 . 2 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
2 dmeq 5917 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
32reseq2d 6000 . . . 4 (𝑟 = 𝑅 → ( I ↾ dom 𝑟) = ( I ↾ dom 𝑅))
4 id 22 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
53, 4sseq12d 4029 . . 3 (𝑟 = 𝑅 → (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
6 cnveq 5887 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
76, 4sseq12d 4029 . . 3 (𝑟 = 𝑅 → (𝑟𝑟𝑅𝑅))
85, 7anbi12d 632 . 2 (𝑟 = 𝑅 → ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ↔ (( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅)))
91, 8rabeqel 38236 1 (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  cin 3962  wss 3963   I cid 5582  ccnv 5688  dom cdm 5689  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:  elrefsymrels3  38552
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