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Theorem elrefrels3 38047
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 38042 . 2 RefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦)}
2 dmeq 5900 . . 3 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3 rneq 5932 . . . 4 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
4 breq 5145 . . . . 5 (𝑟 = 𝑅 → (𝑥𝑟𝑦𝑥𝑅𝑦))
54imbi2d 339 . . . 4 (𝑟 = 𝑅 → ((𝑥 = 𝑦𝑥𝑟𝑦) ↔ (𝑥 = 𝑦𝑥𝑅𝑦)))
63, 5raleqbidv 3330 . . 3 (𝑟 = 𝑅 → (∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦) ↔ ∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦)))
72, 6raleqbidv 3330 . 2 (𝑟 = 𝑅 → (∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦) ↔ ∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦)))
81, 7rabeqel 37782 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 394   = wceq 1533  wcel 2098  wral 3051   class class class wbr 5143  dom cdm 5672  ran crn 5673   Rels crels 37707   RefRels crefrels 37710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-rels 38013  df-ssr 38026  df-refs 38038  df-refrels 38039
This theorem is referenced by: (None)
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