Theorem dfeqvrels2 37458

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 37454	. . 3 ⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
2		refsymrels2 37435	. . . 4 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
3		dftrrels2 37445	. . . 4 ⊢ TrRels = {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟}
4	2, 3	ineq12i 4211	. . 3 ⊢ (( RefRels ∩ SymRels ) ∩ TrRels ) = ({𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} ∩ {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟})
5		inrab 4307	. . 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 2765	. 2 ⊢ EqvRels = {𝑟 ∈ Rels ∣ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟) ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}
7		df-3an 1090	. . 3 ⊢ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ↔ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟) ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟))
8	7	rabbii 3439	. 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 2764	1 ⊢ EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}

Syntax hints:   ∧ wa 397   ∧ w3a 1088   = wceq 1542  {crab 3433   ∩ cin 3948   ⊆ wss 3949   I cid 5574  ^◡ccnv 5676  dom cdm 5677   ↾ cres 5679   ∘ ccom 5681   Rels crels 37045   RefRels crefrels 37048   SymRels csymrels 37054   TrRels ctrrels 37057   EqvRels ceqvrels 37059

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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

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 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-rels 37355  df-ssr 37368  df-refs 37380  df-refrels 37381  df-syms 37412  df-symrels 37413  df-trs 37442  df-trrels 37443  df-eqvrels 37454

This theorem is referenced by:  dfeqvrels3  37459  eleqvrels2  37462