Theorem dfeqvrels2 39171

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 39167	. . 3 ⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
2		refsymrels2 39148	. . . 4 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
3		dftrrels2 39158	. . . 4 ⊢ TrRels = {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟}
4	2, 3	ineq12i 4170	. . 3 ⊢ (( RefRels ∩ SymRels ) ∩ TrRels ) = ({𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} ∩ {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟})
5		inrab 4268	. . 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 2789	. 2 ⊢ EqvRels = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟) ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}
7		df-3an 1100	. . 3 ⊢ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ↔ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟) ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟))
8	7	rabbii 3419	. 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 2788	1 ⊢ EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}

Syntax hints:   ∧ wa 399   ∧ w3a 1098   = wceq 1560  {crab 3414   ∩ cin 3903   ⊆ wss 3904   I cid 5541  ^◡ccnv 5646  dom cdm 5647   ↾ cres 5649   ∘ ccom 5651   Rels crels 38684   RefRels crefrels 38687   SymRels csymrels 38693   TrRels ctrrels 38696   EqvRels ceqvrels 38698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-rels 38939  df-ssr 39077  df-refs 39089  df-refrels 39090  df-syms 39121  df-symrels 39122  df-trs 39155  df-trrels 39156  df-eqvrels 39167

This theorem is referenced by:  dfeqvrels3  39172  eleqvrels2  39175