Theorem eqvrelthi 36653

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 36652	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
4	2, 3	eqvrelth 36651	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
5	1, 4	mpbid 231	1 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Syntax hints:   → wi 4   = wceq 1539   class class class wbr 5070  [cec 8454   EqvRel weqvrel 36277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ec 8458  df-refrel 36557  df-symrel 36585  df-trrel 36615  df-eqvrel 36625

This theorem is referenced by:  eqvreldisj  36654  eqvrelqsel  36656