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

Theorem eqvrelref 36723
Description: An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.)
Hypotheses
Ref Expression
eqvrelref.1 (𝜑 → EqvRel 𝑅)
eqvrelref.2 (𝜑𝐴 ∈ dom 𝑅)
Assertion
Ref Expression
eqvrelref (𝜑𝐴𝑅𝐴)

Proof of Theorem eqvrelref
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqvrelref.2 . . 3 (𝜑𝐴 ∈ dom 𝑅)
2 eqvrelref.1 . . . 4 (𝜑 → EqvRel 𝑅)
3 eqvrelrel 36710 . . . 4 ( EqvRel 𝑅 → Rel 𝑅)
4 releldmb 5855 . . . 4 (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
52, 3, 43syl 18 . . 3 (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
61, 5mpbid 231 . 2 (𝜑 → ∃𝑥 𝐴𝑅𝑥)
72adantr 481 . . 3 ((𝜑𝐴𝑅𝑥) → EqvRel 𝑅)
8 simpr 485 . . 3 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝑥)
97, 8, 8eqvreltr4d 36722 . 2 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝐴)
106, 9exlimddv 1938 1 (𝜑𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wex 1782  wcel 2106   class class class wbr 5074  dom cdm 5589  Rel wrel 5594   EqvRel weqvrel 36350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-refrel 36630  df-symrel 36658  df-trrel 36688  df-eqvrel 36698
This theorem is referenced by:  eqvrelth  36724
  Copyright terms: Public domain W3C validator