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Theorem eqvrelcl 34991
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 34976 . . 3 ( EqvRel 𝑅 → Rel 𝑅)
31, 2syl 17 . 2 (𝜑 → Rel 𝑅)
4 eqvrelcl.2 . 2 (𝜑𝐴𝑅𝐵)
5 releldm 5606 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
63, 4, 5syl2anc 579 1 (𝜑𝐴 ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107   class class class wbr 4888  dom cdm 5357  Rel wrel 5362   EqvRel weqvrel 34632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-refrel 34899  df-symrel 34927  df-trrel 34957  df-eqvrel 34967
This theorem is referenced by:  eqvrelthi  34992
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