Theorem ecss 8755

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 8705	. . 3 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
2		imassrn 6078	. . 3 ⊢ (𝑅 “ {𝐴}) ⊆ ran 𝑅
3	1, 2	eqsstri 3986	. 2 ⊢ [𝐴]𝑅 ⊆ ran 𝑅
4		ecss.1	. . 3 ⊢ (𝜑 → 𝑅 Er 𝑋)
5		errn 8726	. . 3 ⊢ (𝑅 Er 𝑋 → ran 𝑅 = 𝑋)
6	4, 5	syl 18	. 2 ⊢ (𝜑 → ran 𝑅 = 𝑋)
7	3, 6	sseqtrid 3982	1 ⊢ (𝜑 → [𝐴]𝑅 ⊆ 𝑋)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3908  {csn 4594  ran crn 5667   “ cima 5669   Er wer 8700  [cec 8701

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

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 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-er 8703  df-ec 8705

This theorem is used by:  qsss  8782  divsfval  17626  ghmqusnsglem1  19381  ghmquskerlem1  19384  sylow1lem5  19703  sylow2alem2  19719  sylow2blem1  19721  sylow3lem3  19730  vitalilem2  25805  qsalrel  43050