Theorem eqvrelthi 38615

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 38614	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
4	2, 3	eqvrelth 38613	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
5	1, 4	mpbid 232	1 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Syntax hints:   → wi 4   = wceq 1539   class class class wbr 5142  [cec 8744   EqvRel weqvrel 38200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ec 8748  df-refrel 38514  df-symrel 38546  df-trrel 38576  df-eqvrel 38587

This theorem is referenced by:  eqvreldisj  38616  eqvrelqsel  38618