Theorem eqvrelthi 36289

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 36288	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
4	2, 3	eqvrelth 36287	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
5	1, 4	mpbid 235	1 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Syntax hints:   → wi 4   = wceq 1539   class class class wbr 5033  [cec 8298   EqvRel weqvrel 35911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ec 8302  df-refrel 36193  df-symrel 36221  df-trrel 36251  df-eqvrel 36261

This theorem is referenced by:  eqvreldisj  36290  eqvrelqsel  36292