Theorem elrefrels2 38509

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 38504	. 2 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2		dmeq 5867	. . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3		rneq 5900	. . . . 5 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
4	2, 3	xpeq12d 5669	. . . 4 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅))
5	4	ineq2d 4183	. . 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 3980	. 2 ⊢ (𝑟 = 𝑅 → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅))
8	1, 7	rabeqel 38243	1 ⊢ (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3913   ⊆ wss 3914   I cid 5532   × cxp 5636  dom cdm 5638  ran crn 5639   Rels crels 38171   RefRels crefrels 38174

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-rels 38476  df-ssr 38489  df-refs 38501  df-refrels 38502

This theorem is referenced by:  elrefrelsrel  38511