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Theorem elrefrels2 36912
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 36907 . 2 RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2 dmeq 5857 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3 rneq 5889 . . . . 5 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
42, 3xpeq12d 5662 . . . 4 (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅))
54ineq2d 4170 . . 3 (𝑟 = 𝑅 → ( I ∩ (dom 𝑟 × ran 𝑟)) = ( I ∩ (dom 𝑅 × ran 𝑅)))
6 id 22 . . 3 (𝑟 = 𝑅𝑟 = 𝑅)
75, 6sseq12d 3975 . 2 (𝑟 = 𝑅 → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅))
81, 7rabeqel 36646 1 (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  cin 3907  wss 3908   I cid 5528   × cxp 5629  dom cdm 5631  ran crn 5632   Rels crels 36568   RefRels crefrels 36571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-rels 36879  df-ssr 36892  df-refs 36904  df-refrels 36905
This theorem is referenced by:  elrefrelsrel  36914
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