Theorem ecss 8688

Description: An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
ecss.1	⊢ (𝜑 → 𝑅 Er 𝑋)

Assertion

Ref	Expression
ecss	⊢ (𝜑 → [𝐴]𝑅 ⊆ 𝑋)

Proof of Theorem ecss

Step	Hyp	Ref	Expression
1		df-ec 8638	. . 3 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
2		imassrn 6030	. . 3 ⊢ (𝑅 “ {𝐴}) ⊆ ran 𝑅
3	1, 2	eqsstri 3969	. 2 ⊢ [𝐴]𝑅 ⊆ ran 𝑅
4		ecss.1	. . 3 ⊢ (𝜑 → 𝑅 Er 𝑋)
5		errn 8659	. . 3 ⊢ (𝑅 Er 𝑋 → ran 𝑅 = 𝑋)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → ran 𝑅 = 𝑋)
7	3, 6	sseqtrid 3965	1 ⊢ (𝜑 → [𝐴]𝑅 ⊆ 𝑋)

Syntax hints:   → wi 4   = wceq 1542   ⊆ wss 3890  {csn 4568  ran crn 5625   “ cima 5627   Er wer 8633  [cec 8634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-er 8636  df-ec 8638

This theorem is referenced by:  qsss  8715  divsfval  17502  ghmqusnsglem1  19246  ghmquskerlem1  19249  sylow1lem5  19568  sylow2alem2  19584  sylow2blem1  19586  sylow3lem3  19595  vitalilem2  25586  qsalrel  42694