Theorem elcnvrefrels3 39292

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 39286	. 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)}
2		dmeq 5892	. . 3 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3		rneq 5925	. . . 4 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
4		breq 5110	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑦 ↔ 𝑥𝑅𝑦))
5	4	imbi1d 344	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑥𝑟𝑦 → 𝑥 = 𝑦) ↔ (𝑥𝑅𝑦 → 𝑥 = 𝑦)))
6	3, 5	raleqbidv 3337	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦)))
7	2, 6	raleqbidv 3337	. 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 38934	1 ⊢ (𝑅 ∈ CnvRefRels ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ 𝑅 ∈ Rels ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078   class class class wbr 5108  dom cdm 5660  ran crn 5661   Rels crels 38862   CnvRefRels ccnvrefrels 38868

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-dm 5670  df-rn 5671  df-ssr 39255  df-cnvrefs 39282  df-cnvrefrels 39283

This theorem is used by: (None)