Theorem elcnvrefrels3 39461

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 39455	. 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)}
2		dmeq 5882	. . 3 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3		rneq 5915	. . . 4 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
4		breq 5105	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑦 ↔ 𝑥𝑅𝑦))
5	4	imbi1d 344	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑥𝑟𝑦 → 𝑥 = 𝑦) ↔ (𝑥𝑅𝑦 → 𝑥 = 𝑦)))
6	3, 5	raleqbidv 3334	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦)))
7	2, 6	raleqbidv 3334	. 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 39103	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 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   class class class wbr 5103  dom cdm 5648  ran crn 5649   Rels crels 39031   CnvRefRels ccnvrefrels 39037

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5249  ax-pow 5327  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-dm 5658  df-rn 5659  df-ssr 39424  df-cnvrefs 39451  df-cnvrefrels 39452

This theorem is used by: (None)