Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eleqvrels3 Structured version   Visualization version   GIF version

Theorem eleqvrels3 36139
 Description: Element of the class of equivalence relations. (Contributed by Peter Mazsa, 24-Aug-2021.)
Assertion
Ref Expression
eleqvrels3 (𝑅 ∈ EqvRels ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ 𝑅 ∈ Rels ))
Distinct variable group:   𝑥,𝑅,𝑦,𝑧

Proof of Theorem eleqvrels3
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dfeqvrels3 36135 . 2 EqvRels = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
2 dmeq 5742 . . . 4 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3 breq 5036 . . . 4 (𝑟 = 𝑅 → (𝑥𝑟𝑥𝑥𝑅𝑥))
42, 3raleqbidv 3355 . . 3 (𝑟 = 𝑅 → (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥))
5 breq 5036 . . . . 5 (𝑟 = 𝑅 → (𝑥𝑟𝑦𝑥𝑅𝑦))
6 breq 5036 . . . . 5 (𝑟 = 𝑅 → (𝑦𝑟𝑥𝑦𝑅𝑥))
75, 6imbi12d 348 . . . 4 (𝑟 = 𝑅 → ((𝑥𝑟𝑦𝑦𝑟𝑥) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥)))
872albidv 1924 . . 3 (𝑟 = 𝑅 → (∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
9 breq 5036 . . . . . . 7 (𝑟 = 𝑅 → (𝑦𝑟𝑧𝑦𝑅𝑧))
105, 9anbi12d 633 . . . . . 6 (𝑟 = 𝑅 → ((𝑥𝑟𝑦𝑦𝑟𝑧) ↔ (𝑥𝑅𝑦𝑦𝑅𝑧)))
11 breq 5036 . . . . . 6 (𝑟 = 𝑅 → (𝑥𝑟𝑧𝑥𝑅𝑧))
1210, 11imbi12d 348 . . . . 5 (𝑟 = 𝑅 → (((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
13122albidv 1924 . . . 4 (𝑟 = 𝑅 → (∀𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
1413albidv 1921 . . 3 (𝑟 = 𝑅 → (∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
154, 8, 143anbi123d 1433 . 2 (𝑟 = 𝑅 → ((∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ (∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
161, 15rabeqel 35827 1 (𝑅 ∈ EqvRels ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ 𝑅 ∈ Rels ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∀wral 3106   class class class wbr 5034  dom cdm 5523   Rels crels 35766   EqvRels ceqvrels 35780 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-rels 36036  df-ssr 36049  df-refs 36061  df-refrels 36062  df-syms 36089  df-symrels 36090  df-trs 36119  df-trrels 36120  df-eqvrels 36130 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator