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Theorem elcnvrefrels3 35998
 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 35994 . 2 CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦)}
2 dmeq 5741 . . 3 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3 rneq 5775 . . . 4 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
4 breq 5035 . . . . 5 (𝑟 = 𝑅 → (𝑥𝑟𝑦𝑥𝑅𝑦))
54imbi1d 345 . . . 4 (𝑟 = 𝑅 → ((𝑥𝑟𝑦𝑥 = 𝑦) ↔ (𝑥𝑅𝑦𝑥 = 𝑦)))
63, 5raleqbidv 3354 . . 3 (𝑟 = 𝑅 → (∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦) ↔ ∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦𝑥 = 𝑦)))
72, 6raleqbidv 3354 . 2 (𝑟 = 𝑅 → (∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦) ↔ ∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥𝑅𝑦𝑥 = 𝑦)))
81, 7rabeqel 35743 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 1538   ∈ wcel 2111  ∀wral 3106   class class class wbr 5033  dom cdm 5522  ran crn 5523   Rels crels 35682   CnvRefRels ccnvrefrels 35688 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 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451 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 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-dm 5532  df-rn 5533  df-ssr 35965  df-cnvrefs 35990  df-cnvrefrels 35991 This theorem is referenced by: (None)
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