Theorem eqvrelref 38138

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 38125	. . . 4 ⊢ ( EqvRel 𝑅 → Rel 𝑅)
4		releldmb 5942	. . . 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 231	. 2 ⊢ (𝜑 → ∃𝑥 𝐴𝑅𝑥)
7	2	adantr 479	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → EqvRel 𝑅)
8		simpr 483	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝑥)
9	7, 8, 8	eqvreltr4d 38137	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝐴)
10	6, 9	exlimddv 1930	1 ⊢ (𝜑 → 𝐴𝑅𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  ∃wex 1773   ∈ wcel 2098   class class class wbr 5143  dom cdm 5672  Rel wrel 5677   EqvRel weqvrel 37722

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 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-refrel 38040  df-symrel 38072  df-trrel 38102  df-eqvrel 38113

This theorem is referenced by:  eqvrelth  38139