Theorem eqvrelref 38863

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 38850	. . . 4 ⊢ ( EqvRel 𝑅 → Rel 𝑅)
4		releldmb 5895	. . . 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 38862	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝐴)
10	6, 9	exlimddv 1936	1 ⊢ (𝜑 → 𝐴𝑅𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1780   ∈ wcel 2113   class class class wbr 5098  dom cdm 5624  Rel wrel 5629   EqvRel weqvrel 38396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-refrel 38761  df-symrel 38793  df-trrel 38827  df-eqvrel 38838

This theorem is referenced by:  eqvrelth  38864