Theorem elrefrels3 38934

Description: Element of the class of reflexive relations. (Contributed by Peter Mazsa, 23-Jul-2019.)

Assertion

Ref	Expression
elrefrels3	⊢ (𝑅 ∈ RefRels ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) ∧ 𝑅 ∈ Rels ))

Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elrefrels3

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrefrels3 38929	. 2 ⊢ RefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦)}
2		dmeq 5852	. . 3 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3		rneq 5885	. . . 4 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
4		breq 5088	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑦 ↔ 𝑥𝑅𝑦))
5	4	imbi2d 340	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑥 = 𝑦 → 𝑥𝑟𝑦) ↔ (𝑥 = 𝑦 → 𝑥𝑅𝑦)))
6	3, 5	raleqbidv 3312	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦) ↔ ∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦)))
7	2, 6	raleqbidv 3312	. 2 ⊢ (𝑟 = 𝑅 → (∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦) ↔ ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦)))
8	1, 7	rabeqel 38592	1 ⊢ (𝑅 ∈ RefRels ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) ∧ 𝑅 ∈ Rels ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5086  dom cdm 5624  ran crn 5625   Rels crels 38520   RefRels crefrels 38523

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-rels 38775  df-ssr 38913  df-refs 38925  df-refrels 38926

This theorem is referenced by: (None)