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Theorem eleqvrels3 38121
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 38117 . 2 EqvRels = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
2 dmeq 5900 . . . 4 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3 breq 5145 . . . 4 (𝑟 = 𝑅 → (𝑥𝑟𝑥𝑥𝑅𝑥))
42, 3raleqbidv 3330 . . 3 (𝑟 = 𝑅 → (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥))
5 breq 5145 . . . . 5 (𝑟 = 𝑅 → (𝑥𝑟𝑦𝑥𝑅𝑦))
6 breq 5145 . . . . 5 (𝑟 = 𝑅 → (𝑦𝑟𝑥𝑦𝑅𝑥))
75, 6imbi12d 343 . . . 4 (𝑟 = 𝑅 → ((𝑥𝑟𝑦𝑦𝑟𝑥) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥)))
872albidv 1918 . . 3 (𝑟 = 𝑅 → (∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
9 breq 5145 . . . . . . 7 (𝑟 = 𝑅 → (𝑦𝑟𝑧𝑦𝑅𝑧))
105, 9anbi12d 630 . . . . . 6 (𝑟 = 𝑅 → ((𝑥𝑟𝑦𝑦𝑟𝑧) ↔ (𝑥𝑅𝑦𝑦𝑅𝑧)))
11 breq 5145 . . . . . 6 (𝑟 = 𝑅 → (𝑥𝑟𝑧𝑥𝑅𝑧))
1210, 11imbi12d 343 . . . . 5 (𝑟 = 𝑅 → (((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
13122albidv 1918 . . . 4 (𝑟 = 𝑅 → (∀𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
1413albidv 1915 . . 3 (𝑟 = 𝑅 → (∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
154, 8, 143anbi123d 1432 . 2 (𝑟 = 𝑅 → ((∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ (∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
161, 15rabeqel 37782 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 205  wa 394  w3a 1084  wal 1531   = wceq 1533  wcel 2098  wral 3051   class class class wbr 5143  dom cdm 5672   Rels crels 37707   EqvRels ceqvrels 37721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-rels 38013  df-ssr 38026  df-refs 38038  df-refrels 38039  df-syms 38070  df-symrels 38071  df-trs 38100  df-trrels 38101  df-eqvrels 38112
This theorem is referenced by: (None)
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