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Theorem eqvrelcl 36023
 Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.)
Hypotheses
Ref Expression
eqvrelcl.1 (𝜑 → EqvRel 𝑅)
eqvrelcl.2 (𝜑𝐴𝑅𝐵)
Assertion
Ref Expression
eqvrelcl (𝜑𝐴 ∈ dom 𝑅)

Proof of Theorem eqvrelcl
StepHypRef Expression
1 eqvrelcl.1 . . 3 (𝜑 → EqvRel 𝑅)
2 eqvrelrel 36008 . . 3 ( EqvRel 𝑅 → Rel 𝑅)
31, 2syl 17 . 2 (𝜑 → Rel 𝑅)
4 eqvrelcl.2 . 2 (𝜑𝐴𝑅𝐵)
5 releldm 5778 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
63, 4, 5syl2anc 587 1 (𝜑𝐴 ∈ dom 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111   class class class wbr 5030  dom cdm 5519  Rel wrel 5524   EqvRel weqvrel 35646 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 5167  ax-nul 5174  ax-pr 5295 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 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-refrel 35928  df-symrel 35956  df-trrel 35986  df-eqvrel 35996 This theorem is referenced by:  eqvrelthi  36024  erim2  36087
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