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Theorem elrefrels2 38500
Description: Element of the class of reflexive relations. (Contributed by Peter Mazsa, 23-Jul-2019.)
Assertion
Ref Expression
elrefrels2 (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅 ∈ Rels ))

Proof of Theorem elrefrels2
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dfrefrels2 38495 . 2 RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2 dmeq 5917 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3 rneq 5950 . . . . 5 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
42, 3xpeq12d 5720 . . . 4 (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅))
54ineq2d 4228 . . 3 (𝑟 = 𝑅 → ( I ∩ (dom 𝑟 × ran 𝑟)) = ( I ∩ (dom 𝑅 × ran 𝑅)))
6 id 22 . . 3 (𝑟 = 𝑅𝑟 = 𝑅)
75, 6sseq12d 4029 . 2 (𝑟 = 𝑅 → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅))
81, 7rabeqel 38236 1 (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  cin 3962  wss 3963   I cid 5582   × cxp 5687  dom cdm 5689  ran crn 5690   Rels crels 38164   RefRels crefrels 38167
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-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
This theorem is referenced by:  elrefrelsrel  38502
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