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Theorem dfeqvrels3 34827
Description: Alternate definition of the class of equivalence relations. (Contributed by Peter Mazsa, 2-Dec-2019.)
Assertion
Ref Expression
dfeqvrels3 EqvRels = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
Distinct variable group:   𝑥,𝑟,𝑦,𝑧

Proof of Theorem dfeqvrels3
StepHypRef Expression
1 dfeqvrels2 34826 . 2 EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}
2 idrefALT 5726 . . 3 (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥)
3 cnvsym 5728 . . 3 (𝑟𝑟 ↔ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥))
4 cotr 5725 . . 3 ((𝑟𝑟) ⊆ 𝑟 ↔ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))
52, 3, 43anbi123i 1195 . 2 ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)))
61, 5rabbieq 34515 1 EqvRels = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108  wal 1651   = wceq 1653  wral 3089  {crab 3093  wss 3769   class class class wbr 4843   I cid 5219  ccnv 5311  dom cdm 5312  cres 5314  ccom 5316   Rels crels 34471   EqvRels ceqvrels 34485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-rels 34729  df-ssr 34742  df-refs 34754  df-refrels 34755  df-syms 34782  df-symrels 34783  df-trs 34812  df-trrels 34813  df-eqvrels 34823
This theorem is referenced by:  eleqvrels3  34831
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