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Theorem eleqvrels3 35708
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 35704 . 2 EqvRels = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
2 dmeq 5765 . . . 4 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3 breq 5059 . . . 4 (𝑟 = 𝑅 → (𝑥𝑟𝑥𝑥𝑅𝑥))
42, 3raleqbidv 3399 . . 3 (𝑟 = 𝑅 → (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥))
5 breq 5059 . . . . 5 (𝑟 = 𝑅 → (𝑥𝑟𝑦𝑥𝑅𝑦))
6 breq 5059 . . . . 5 (𝑟 = 𝑅 → (𝑦𝑟𝑥𝑦𝑅𝑥))
75, 6imbi12d 346 . . . 4 (𝑟 = 𝑅 → ((𝑥𝑟𝑦𝑦𝑟𝑥) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥)))
872albidv 1915 . . 3 (𝑟 = 𝑅 → (∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
9 breq 5059 . . . . . . 7 (𝑟 = 𝑅 → (𝑦𝑟𝑧𝑦𝑅𝑧))
105, 9anbi12d 630 . . . . . 6 (𝑟 = 𝑅 → ((𝑥𝑟𝑦𝑦𝑟𝑧) ↔ (𝑥𝑅𝑦𝑦𝑅𝑧)))
11 breq 5059 . . . . . 6 (𝑟 = 𝑅 → (𝑥𝑟𝑧𝑥𝑅𝑧))
1210, 11imbi12d 346 . . . . 5 (𝑟 = 𝑅 → (((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
13122albidv 1915 . . . 4 (𝑟 = 𝑅 → (∀𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
1413albidv 1912 . . 3 (𝑟 = 𝑅 → (∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
154, 8, 143anbi123d 1427 . 2 (𝑟 = 𝑅 → ((∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ (∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
161, 15rabeqel 35397 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 207  wa 396  w3a 1079  wal 1526   = wceq 1528  wcel 2105  wral 3135   class class class wbr 5057  dom cdm 5548   Rels crels 35336   EqvRels ceqvrels 35350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-rels 35605  df-ssr 35618  df-refs 35630  df-refrels 35631  df-syms 35658  df-symrels 35659  df-trs 35688  df-trrels 35689  df-eqvrels 35699
This theorem is referenced by: (None)
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