Theorem dfeqvrels2 38611

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) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}

Proof of Theorem dfeqvrels2

Step	Hyp	Ref	Expression
1		df-eqvrels 38607	. . 3 ⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
2		refsymrels2 38588	. . . 4 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
3		dftrrels2 38598	. . . 4 ⊢ TrRels = {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟}
4	2, 3	ineq12i 4198	. . 3 ⊢ (( RefRels ∩ SymRels ) ∩ TrRels ) = ({𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} ∩ {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟})
5		inrab 4296	. . 3 ⊢ ({𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} ∩ {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟}) = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟) ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}
6	1, 4, 5	3eqtri 2763	. 2 ⊢ EqvRels = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟) ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}
7		df-3an 1088	. . 3 ⊢ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ↔ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟) ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟))
8	7	rabbii 3426	. 2 ⊢ {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟) ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}
9	6, 8	eqtr4i 2762	1 ⊢ EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}

Syntax hints:   ∧ wa 395   ∧ w3a 1086   = wceq 1540  {crab 3420   ∩ cin 3930   ⊆ wss 3931   I cid 5552  ^◡ccnv 5658  dom cdm 5659   ↾ cres 5661   ∘ ccom 5663   Rels crels 38206   RefRels crefrels 38209   SymRels csymrels 38215   TrRels ctrrels 38218   EqvRels ceqvrels 38220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-rels 38508  df-ssr 38521  df-refs 38533  df-refrels 38534  df-syms 38565  df-symrels 38566  df-trs 38595  df-trrels 38596  df-eqvrels 38607

This theorem is referenced by:  dfeqvrels3  38612  eleqvrels2  38615