Theorem dfrefrels3 38537

Description: Alternate definition of the class of reflexive relations. (Contributed by Peter Mazsa, 8-Jul-2019.)

Assertion

Ref	Expression
dfrefrels3	⊢ RefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦)}

Distinct variable group:   𝑥,𝑟,𝑦

Proof of Theorem dfrefrels3

Step	Hyp	Ref	Expression
1		dfrefrels2 38536	. 2 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2		idinxpss 38335	. 2 ⊢ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦))
3	1, 2	rabbieq 3429	1 ⊢ RefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦)}

Syntax hints:   → wi 4   = wceq 1540  ∀wral 3052  {crab 3420   ∩ cin 3930   ⊆ wss 3931   class class class wbr 5124   I cid 5552   × cxp 5657  dom cdm 5659  ran crn 5660   Rels crels 38206   RefRels crefrels 38209

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-rels 38508  df-ssr 38521  df-refs 38533  df-refrels 38534

This theorem is referenced by:  elrefrels3  38542