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Theorem elrefrels3 34583
Description: Element of the class of all reflexive relations. (Contributed by Peter Mazsa, 23-Jul-2019.)
Assertion
Ref Expression
elrefrels3 (𝑅 ∈ RefRels ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ 𝑅 ∈ Rels ))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elrefrels3
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dfrefrels3 34579 . 2 RefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦)}
2 dmeq 5471 . . 3 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3 rneq 5498 . . . 4 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
4 breq 4798 . . . . 5 (𝑟 = 𝑅 → (𝑥𝑟𝑦𝑥𝑅𝑦))
54imbi2d 329 . . . 4 (𝑟 = 𝑅 → ((𝑥 = 𝑦𝑥𝑟𝑦) ↔ (𝑥 = 𝑦𝑥𝑅𝑦)))
63, 5raleqbidv 3283 . . 3 (𝑟 = 𝑅 → (∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦) ↔ ∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦)))
72, 6raleqbidv 3283 . 2 (𝑟 = 𝑅 → (∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦) ↔ ∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦)))
81, 7rabeqel 34335 1 (𝑅 ∈ RefRels ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  wral 3042   class class class wbr 4796  dom cdm 5258  ran crn 5259   Rels crels 34290   RefRels crefrels 34293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-dm 5268  df-rn 5269  df-res 5270  df-rels 34550  df-ssr 34563  df-refs 34575  df-refrels 34576
This theorem is referenced by: (None)
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