Theorem eqvrelref 38633

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 38620	. . . 4 ⊢ ( EqvRel 𝑅 → Rel 𝑅)
4		releldmb 5931	. . . 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 38632	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝐴)
10	6, 9	exlimddv 1935	1 ⊢ (𝜑 → 𝐴𝑅𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1779   ∈ wcel 2109   class class class wbr 5124  dom cdm 5659  Rel wrel 5664   EqvRel weqvrel 38221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-refrel 38535  df-symrel 38567  df-trrel 38597  df-eqvrel 38608

This theorem is referenced by:  eqvrelth  38634