Theorem elrefrels2 38555

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 38550	. 2 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2		dmeq 5838	. . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3		rneq 5871	. . . . 5 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
4	2, 3	xpeq12d 5642	. . . 4 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅))
5	4	ineq2d 4165	. . 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 3963	. 2 ⊢ (𝑟 = 𝑅 → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅))
8	1, 7	rabeqel 38289	1 ⊢ (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ∩ cin 3896   ⊆ wss 3897   I cid 5505   × cxp 5609  dom cdm 5611  ran crn 5612   Rels crels 38217   RefRels crefrels 38220

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-xp 5617  df-rel 5618  df-cnv 5619  df-dm 5621  df-rn 5622  df-res 5623  df-rels 38522  df-ssr 38535  df-refs 38547  df-refrels 38548

This theorem is referenced by:  elrefrelsrel  38557