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 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ 𝑅 ∈ Rels ))

Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elcnvrefrels3

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcnvrefrels3 36382	. 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 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 → (𝑥𝑟𝑦 ↔ 𝑥𝑅𝑦))
5	4	imbi1d 345	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑥𝑟𝑦 → 𝑥 = 𝑦) ↔ (𝑥𝑅𝑦 → 𝑥 = 𝑦)))
6	3, 5	raleqbidv 3313	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦)))
7	2, 6	raleqbidv 3313	. 2 ⊢ (𝑟 = 𝑅 → (∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦)))
8	1, 7	rabeqel 36131	1 ⊢ (𝑅 ∈ CnvRefRels ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ 𝑅 ∈ Rels ))

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)