Theorem eqvrelthi 39895

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 39894	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
4	2, 3	eqvrelth 39893	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
5	1, 4	mpbid 235	1 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   class class class wbr 5108  [cec 8753   EqvRel weqvrel 39398

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ec 8757  df-refrel 39790  df-symrel 39822  df-trrel 39856  df-eqvrel 39867

This theorem is used by:  eqvreldisj  39896  eqvrelqsel  39898