Theorem elcnvrefrels3 35786

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 35782	. 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 5806	. . . 4 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
4		breq 5068	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑦 ↔ 𝑥𝑅𝑦))
5	4	imbi1d 344	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑥𝑟𝑦 → 𝑥 = 𝑦) ↔ (𝑥𝑅𝑦 → 𝑥 = 𝑦)))
6	3, 5	raleqbidv 3401	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦)))
7	2, 6	raleqbidv 3401	. 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 35531	1 ⊢ (𝑅 ∈ CnvRefRels ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ 𝑅 ∈ Rels ))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138   class class class wbr 5066  dom cdm 5555  ran crn 5556   Rels crels 35470   CnvRefRels ccnvrefrels 35476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-dm 5565  df-rn 5566  df-ssr 35753  df-cnvrefs 35778  df-cnvrefrels 35779

This theorem is referenced by: (None)