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Theorem elrefrels2 35917
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 35913 . 2 RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2 dmeq 5736 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3 rneq 5770 . . . . 5 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
42, 3xpeq12d 5550 . . . 4 (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅))
54ineq2d 4139 . . 3 (𝑟 = 𝑅 → ( I ∩ (dom 𝑟 × ran 𝑟)) = ( I ∩ (dom 𝑅 × ran 𝑅)))
6 id 22 . . 3 (𝑟 = 𝑅𝑟 = 𝑅)
75, 6sseq12d 3948 . 2 (𝑟 = 𝑅 → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅))
81, 7rabeqel 35676 1 (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2111  cin 3880  wss 3881   I cid 5424   × cxp 5517  dom cdm 5519  ran crn 5520   Rels crels 35615   RefRels crefrels 35618
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-rels 35885  df-ssr 35898  df-refs 35910  df-refrels 35911
This theorem is referenced by:  elrefrelsrel  35919
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