Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqvrelcl Structured version   Visualization version   GIF version

Theorem eqvrelcl 37995
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 37980 . . 3 ( EqvRel 𝑅 → Rel 𝑅)
31, 2syl 17 . 2 (𝜑 → Rel 𝑅)
4 eqvrelcl.2 . 2 (𝜑𝐴𝑅𝐵)
5 releldm 5937 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
63, 4, 5syl2anc 583 1 (𝜑𝐴 ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098   class class class wbr 5141  dom cdm 5669  Rel wrel 5674   EqvRel weqvrel 37573
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-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-refrel 37895  df-symrel 37927  df-trrel 37957  df-eqvrel 37968
This theorem is referenced by:  eqvrelthi  37996  erimeq2  38061
  Copyright terms: Public domain W3C validator