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Theorem elrefrels3 37027
Description: Element of the class of 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 37022 . 2 RefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦)}
2 dmeq 5860 . . 3 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3 rneq 5892 . . . 4 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
4 breq 5108 . . . . 5 (𝑟 = 𝑅 → (𝑥𝑟𝑦𝑥𝑅𝑦))
54imbi2d 341 . . . 4 (𝑟 = 𝑅 → ((𝑥 = 𝑦𝑥𝑟𝑦) ↔ (𝑥 = 𝑦𝑥𝑅𝑦)))
63, 5raleqbidv 3318 . . 3 (𝑟 = 𝑅 → (∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦) ↔ ∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦)))
72, 6raleqbidv 3318 . 2 (𝑟 = 𝑅 → (∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦) ↔ ∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦)))
81, 7rabeqel 36760 1 (𝑅 ∈ RefRels ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061   class class class wbr 5106  dom cdm 5634  ran crn 5635   Rels crels 36682   RefRels crefrels 36685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-rels 36993  df-ssr 37006  df-refs 37018  df-refrels 37019
This theorem is referenced by: (None)
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