Theorem eqvrelref 38942

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	⊢ (𝜑 → 𝐴 ∈ dom 𝑅)

Assertion

Ref	Expression
eqvrelref	⊢ (𝜑 → 𝐴𝑅𝐴)

Proof of Theorem eqvrelref

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqvrelref.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
2		eqvrelref.1	. . . 4 ⊢ (𝜑 → EqvRel 𝑅)
3		eqvrelrel 38929	. . . 4 ⊢ ( EqvRel 𝑅 → Rel 𝑅)
4		releldmb 5903	. . . 4 ⊢ (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
5	2, 3, 4	3syl 18	. . 3 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
6	1, 5	mpbid 232	. 2 ⊢ (𝜑 → ∃𝑥 𝐴𝑅𝑥)
7	2	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → EqvRel 𝑅)
8		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝑥)
9	7, 8, 8	eqvreltr4d 38941	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝐴)
10	6, 9	exlimddv 1937	1 ⊢ (𝜑 → 𝐴𝑅𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1781   ∈ wcel 2114   class class class wbr 5100  dom cdm 5632  Rel wrel 5637   EqvRel weqvrel 38448

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-refrel 38840  df-symrel 38872  df-trrel 38906  df-eqvrel 38917

This theorem is referenced by:  eqvrelth  38943