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Theorem eqvrelcoss3 35847
Description: Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 28-Apr-2019.)
Assertion
Ref Expression
eqvrelcoss3 ( EqvRel ≀ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Distinct variable group:   𝑥,𝑅,𝑦,𝑧

Proof of Theorem eqvrelcoss3
StepHypRef Expression
1 relcoss 35662 . . 3 Rel ≀ 𝑅
21biantru 532 . 2 ((∀𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ((∀𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ Rel ≀ 𝑅))
3 refrelcosslem 35696 . . 3 𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥
4 symrelcoss3 35699 . . . 4 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel ≀ 𝑅)
54simpli 486 . . 3 𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)
63, 5triantru3 35494 . 2 (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ (∀𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
7 dfeqvrel3 35820 . 2 ( EqvRel ≀ 𝑅 ↔ ((∀𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ Rel ≀ 𝑅))
82, 6, 73bitr4ri 306 1 ( EqvRel ≀ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083  wal 1531  wral 3138   class class class wbr 5058  dom cdm 5549  Rel wrel 5554  ccoss 35447   EqvRel weqvrel 35464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-coss 35653  df-refrel 35746  df-symrel 35774  df-trrel 35804  df-eqvrel 35814
This theorem is referenced by:  eqvrelcoss2  35848  eqvrelcoss4  35849
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