Theorem ecss 8797

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 8747	. . 3 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
2		imassrn 6210	. . 3 ⊢ (𝑅 “ {𝐴}) ⊆ ran 𝑅
3	1, 2	eqsstri 3959	. 2 ⊢ [𝐴]𝑅 ⊆ ran 𝑅
4		ecss.1	. . 3 ⊢ (𝜑 → 𝑅 Er 𝑋)
5		errn 8768	. . 3 ⊢ (𝑅 Er 𝑋 → ran 𝑅 = 𝑋)
6	4, 5	syl 18	. 2 ⊢ (𝜑 → ran 𝑅 = 𝑋)
7	3, 6	sseqtrid 3955	1 ⊢ (𝜑 → [𝐴]𝑅 ⊆ 𝑋)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3900  {csn 4588  ran crn 5656   “ cima 5658   Er wer 8742  [cec 8743

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 5252  ax-pr 5395

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-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 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-er 8745  df-ec 8747

This theorem is used by:  qsss  8824  divsfval  17748  ghmqusnsglem1  19524  ghmquskerlem1  19527  sylow1lem5  19846  sylow2alem2  19862  sylow2blem1  19864  sylow3lem3  19873  vitalilem2  25960  qsalrel  43485