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Theorem dfeqvrels2 37079
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 37075 . . 3 EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
2 refsymrels2 37056 . . . 4 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
3 dftrrels2 37066 . . . 4 TrRels = {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟}
42, 3ineq12i 4175 . . 3 (( RefRels ∩ SymRels ) ∩ TrRels ) = ({𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)} ∩ {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟})
5 inrab 4271 . . 3 ({𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)} ∩ {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟}) = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ∧ (𝑟𝑟) ⊆ 𝑟)}
61, 4, 53eqtri 2769 . 2 EqvRels = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ∧ (𝑟𝑟) ⊆ 𝑟)}
7 df-3an 1090 . . 3 ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ∧ (𝑟𝑟) ⊆ 𝑟))
87rabbii 3416 . 2 {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ∧ (𝑟𝑟) ⊆ 𝑟)}
96, 8eqtr4i 2768 1 EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}
Colors of variables: wff setvar class
Syntax hints:  wa 397  w3a 1088   = wceq 1542  {crab 3410  cin 3914  wss 3915   I cid 5535  ccnv 5637  dom cdm 5638  cres 5640  ccom 5642   Rels crels 36665   RefRels crefrels 36668   SymRels csymrels 36674   TrRels ctrrels 36677   EqvRels ceqvrels 36679
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-rels 36976  df-ssr 36989  df-refs 37001  df-refrels 37002  df-syms 37033  df-symrels 37034  df-trs 37063  df-trrels 37064  df-eqvrels 37075
This theorem is referenced by:  dfeqvrels3  37080  eleqvrels2  37083
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