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Theorem dfeqvrels2 36701
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 36697 . . 3 EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
2 refsymrels2 36679 . . . 4 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
3 dftrrels2 36689 . . . 4 TrRels = {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟}
42, 3ineq12i 4144 . . 3 (( RefRels ∩ SymRels ) ∩ TrRels ) = ({𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)} ∩ {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟})
5 inrab 4240 . . 3 ({𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)} ∩ {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟}) = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ∧ (𝑟𝑟) ⊆ 𝑟)}
61, 4, 53eqtri 2770 . 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 3408 . 2 {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ∧ (𝑟𝑟) ⊆ 𝑟)}
96, 8eqtr4i 2769 1 EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}
Colors of variables: wff setvar class
Syntax hints:  wa 396  w3a 1086   = wceq 1539  {crab 3068  cin 3886  wss 3887   I cid 5488  ccnv 5588  dom cdm 5589  cres 5591  ccom 5593   Rels crels 36335   RefRels crefrels 36338   SymRels csymrels 36344   TrRels ctrrels 36347   EqvRels ceqvrels 36349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-rels 36603  df-ssr 36616  df-refs 36628  df-refrels 36629  df-syms 36656  df-symrels 36657  df-trs 36686  df-trrels 36687  df-eqvrels 36697
This theorem is referenced by:  dfeqvrels3  36702  eleqvrels2  36705
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