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Theorem dfrefrels3 37187
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 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦)}
Distinct variable group:   𝑥,𝑟,𝑦

Proof of Theorem dfrefrels3
StepHypRef Expression
1 dfrefrels2 37186 . 2 RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2 idinxpss 36984 . 2 (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦))
31, 2rabbieq 36921 1 RefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wral 3060  {crab 3431  cin 3943  wss 3944   class class class wbr 5141   I cid 5566   × cxp 5667  dom cdm 5669  ran crn 5670   Rels crels 36848   RefRels crefrels 36851
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-rels 37158  df-ssr 37171  df-refs 37183  df-refrels 37184
This theorem is referenced by:  elrefrels3  37192
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