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Theorem dfeqvrels2 38570
Description: Alternate definition of the class of equivalence relations. (Contributed by Peter Mazsa, 2-Dec-2019.)
Assertion
Ref Expression
dfeqvrels2 EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}

Proof of Theorem dfeqvrels2
StepHypRef Expression
1 df-eqvrels 38566 . . 3 EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
2 refsymrels2 38547 . . . 4 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
3 dftrrels2 38557 . . . 4 TrRels = {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟}
42, 3ineq12i 4226 . . 3 (( RefRels ∩ SymRels ) ∩ TrRels ) = ({𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)} ∩ {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟})
5 inrab 4322 . . 3 ({𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)} ∩ {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟}) = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ∧ (𝑟𝑟) ⊆ 𝑟)}
61, 4, 53eqtri 2767 . 2 EqvRels = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ∧ (𝑟𝑟) ⊆ 𝑟)}
7 df-3an 1088 . . 3 ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ∧ (𝑟𝑟) ⊆ 𝑟))
87rabbii 3439 . 2 {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ∧ (𝑟𝑟) ⊆ 𝑟)}
96, 8eqtr4i 2766 1 EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3a 1086   = wceq 1537  {crab 3433  cin 3962  wss 3963   I cid 5582  ccnv 5688  dom cdm 5689  cres 5691  ccom 5693   Rels crels 38164   RefRels crefrels 38167   SymRels csymrels 38173   TrRels ctrrels 38176   EqvRels ceqvrels 38178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-rels 38467  df-ssr 38480  df-refs 38492  df-refrels 38493  df-syms 38524  df-symrels 38525  df-trs 38554  df-trrels 38555  df-eqvrels 38566
This theorem is referenced by:  dfeqvrels3  38571  eleqvrels2  38574
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