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Theorem dfeqvrels2 38544
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 38540 . . 3 EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
2 refsymrels2 38521 . . . 4 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
3 dftrrels2 38531 . . . 4 TrRels = {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟}
42, 3ineq12i 4239 . . 3 (( RefRels ∩ SymRels ) ∩ TrRels ) = ({𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)} ∩ {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟})
5 inrab 4335 . . 3 ({𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)} ∩ {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟}) = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ∧ (𝑟𝑟) ⊆ 𝑟)}
61, 4, 53eqtri 2772 . 2 EqvRels = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ∧ (𝑟𝑟) ⊆ 𝑟)}
7 df-3an 1089 . . 3 ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ∧ (𝑟𝑟) ⊆ 𝑟))
87rabbii 3449 . 2 {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ∧ (𝑟𝑟) ⊆ 𝑟)}
96, 8eqtr4i 2771 1 EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3a 1087   = wceq 1537  {crab 3443  cin 3975  wss 3976   I cid 5592  ccnv 5699  dom cdm 5700  cres 5702  ccom 5704   Rels crels 38137   RefRels crefrels 38140   SymRels csymrels 38146   TrRels ctrrels 38149   EqvRels ceqvrels 38151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-rels 38441  df-ssr 38454  df-refs 38466  df-refrels 38467  df-syms 38498  df-symrels 38499  df-trs 38528  df-trrels 38529  df-eqvrels 38540
This theorem is referenced by:  dfeqvrels3  38545  eleqvrels2  38548
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