Theorem dfeqvrels2 38333

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 38329	. . 3 ⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
2		refsymrels2 38310	. . . 4 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
3		dftrrels2 38320	. . . 4 ⊢ TrRels = {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟}
4	2, 3	ineq12i 4221	. . 3 ⊢ (( RefRels ∩ SymRels ) ∩ TrRels ) = ({𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} ∩ {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟})
5		inrab 4318	. . 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 2761	. 2 ⊢ EqvRels = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟) ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}
7		df-3an 1086	. . 3 ⊢ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ↔ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟) ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟))
8	7	rabbii 3434	. 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 2760	1 ⊢ EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}

Syntax hints:   ∧ wa 394   ∧ w3a 1084   = wceq 1534  {crab 3428   ∩ cin 3958   ⊆ wss 3959   I cid 5583  ^◡ccnv 5685  dom cdm 5686   ↾ cres 5688   ∘ ccom 5690   Rels crels 37925   RefRels crefrels 37928   SymRels csymrels 37934   TrRels ctrrels 37937   EqvRels ceqvrels 37939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700  ax-sep 5307  ax-nul 5314  ax-pr 5437

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-nul 4336  df-if 4537  df-pw 4612  df-sn 4637  df-pr 4639  df-op 4643  df-br 5157  df-opab 5219  df-id 5584  df-xp 5692  df-rel 5693  df-cnv 5694  df-co 5695  df-dm 5696  df-rn 5697  df-res 5698  df-rels 38230  df-ssr 38243  df-refs 38255  df-refrels 38256  df-syms 38287  df-symrels 38288  df-trs 38317  df-trrels 38318  df-eqvrels 38329

This theorem is referenced by:  dfeqvrels3  38334  eleqvrels2  38337