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Theorem dfeqvrels2 37453
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 37449 . . 3 EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
2 refsymrels2 37430 . . . 4 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
3 dftrrels2 37440 . . . 4 TrRels = {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟}
42, 3ineq12i 4210 . . 3 (( RefRels ∩ SymRels ) ∩ TrRels ) = ({𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)} ∩ {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟})
5 inrab 4306 . . 3 ({𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)} ∩ {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟}) = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ∧ (𝑟𝑟) ⊆ 𝑟)}
61, 4, 53eqtri 2764 . 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 3438 . 2 {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ∧ (𝑟𝑟) ⊆ 𝑟)}
96, 8eqtr4i 2763 1 EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}
Colors of variables: wff setvar class
Syntax hints:  wa 396  w3a 1087   = wceq 1541  {crab 3432  cin 3947  wss 3948   I cid 5573  ccnv 5675  dom cdm 5676  cres 5678  ccom 5680   Rels crels 37040   RefRels crefrels 37043   SymRels csymrels 37049   TrRels ctrrels 37052   EqvRels ceqvrels 37054
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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
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 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-rels 37350  df-ssr 37363  df-refs 37375  df-refrels 37376  df-syms 37407  df-symrels 37408  df-trs 37437  df-trrels 37438  df-eqvrels 37449
This theorem is referenced by:  dfeqvrels3  37454  eleqvrels2  37457
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