Theorem eqvrelthi 38649

Description: Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.)

Hypotheses

Ref	Expression
eqvrelthi.1	⊢ (𝜑 → EqvRel 𝑅)
eqvrelthi.2	⊢ (𝜑 → 𝐴𝑅𝐵)

Assertion

Ref	Expression
eqvrelthi	⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Proof of Theorem eqvrelthi

Step	Hyp	Ref	Expression
1		eqvrelthi.2	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
2		eqvrelthi.1	. . 3 ⊢ (𝜑 → EqvRel 𝑅)
3	2, 1	eqvrelcl 38648	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
4	2, 3	eqvrelth 38647	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
5	1, 4	mpbid 232	1 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Syntax hints:   → wi 4   = wceq 1541   class class class wbr 5091  [cec 8620   EqvRel weqvrel 38231

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

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 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-ec 8624  df-refrel 38548  df-symrel 38580  df-trrel 38610  df-eqvrel 38621

This theorem is referenced by:  eqvreldisj  38650  eqvrelqsel  38652