Theorem elrefrels2 38519

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

Assertion

Ref	Expression
elrefrels2	⊢ (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))

Proof of Theorem elrefrels2

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrefrels2 38514	. 2 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2		dmeq 5914	. . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3		rneq 5947	. . . . 5 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
4	2, 3	xpeq12d 5716	. . . 4 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅))
5	4	ineq2d 4220	. . 3 ⊢ (𝑟 = 𝑅 → ( I ∩ (dom 𝑟 × ran 𝑟)) = ( I ∩ (dom 𝑅 × ran 𝑅)))
6		id 22	. . 3 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
7	5, 6	sseq12d 4017	. 2 ⊢ (𝑟 = 𝑅 → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅))
8	1, 7	rabeqel 38255	1 ⊢ (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ∩ cin 3950   ⊆ wss 3951   I cid 5577   × cxp 5683  dom cdm 5685  ran crn 5686   Rels crels 38184   RefRels crefrels 38187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-rels 38486  df-ssr 38499  df-refs 38511  df-refrels 38512

This theorem is referenced by:  elrefrelsrel  38521