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Theorem elcnvrefrels3 36386
Description: Element of the class of converse reflexive relations. (Contributed by Peter Mazsa, 30-Aug-2021.)
Assertion
Ref Expression
elcnvrefrels3 (𝑅 ∈ CnvRefRels ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥𝑅𝑦𝑥 = 𝑦) ∧ 𝑅 ∈ Rels ))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elcnvrefrels3
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dfcnvrefrels3 36382 . 2 CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦)}
2 dmeq 5772 . . 3 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3 rneq 5805 . . . 4 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
4 breq 5055 . . . . 5 (𝑟 = 𝑅 → (𝑥𝑟𝑦𝑥𝑅𝑦))
54imbi1d 345 . . . 4 (𝑟 = 𝑅 → ((𝑥𝑟𝑦𝑥 = 𝑦) ↔ (𝑥𝑅𝑦𝑥 = 𝑦)))
63, 5raleqbidv 3313 . . 3 (𝑟 = 𝑅 → (∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦) ↔ ∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦𝑥 = 𝑦)))
72, 6raleqbidv 3313 . 2 (𝑟 = 𝑅 → (∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦) ↔ ∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥𝑅𝑦𝑥 = 𝑦)))
81, 7rabeqel 36131 1 (𝑅 ∈ CnvRefRels ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥𝑅𝑦𝑥 = 𝑦) ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wral 3061   class class class wbr 5053  dom cdm 5551  ran crn 5552   Rels crels 36072   CnvRefRels ccnvrefrels 36078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-dm 5561  df-rn 5562  df-ssr 36353  df-cnvrefs 36378  df-cnvrefrels 36379
This theorem is referenced by: (None)
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