Theorem ecss 8724

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 8675	. . 3 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
2		imassrn 6044	. . 3 ⊢ (𝑅 “ {𝐴}) ⊆ ran 𝑅
3	1, 2	eqsstri 3995	. 2 ⊢ [𝐴]𝑅 ⊆ ran 𝑅
4		ecss.1	. . 3 ⊢ (𝜑 → 𝑅 Er 𝑋)
5		errn 8695	. . 3 ⊢ (𝑅 Er 𝑋 → ran 𝑅 = 𝑋)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → ran 𝑅 = 𝑋)
7	3, 6	sseqtrid 3991	1 ⊢ (𝜑 → [𝐴]𝑅 ⊆ 𝑋)

Syntax hints:   → wi 4   = wceq 1540   ⊆ wss 3916  {csn 4591  ran crn 5641   “ cima 5643   Er wer 8670  [cec 8671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-xp 5646  df-rel 5647  df-cnv 5648  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-er 8673  df-ec 8675

This theorem is referenced by:  qsss  8751  divsfval  17516  ghmqusnsglem1  19218  ghmquskerlem1  19221  sylow1lem5  19538  sylow2alem2  19554  sylow2blem1  19556  sylow3lem3  19565  vitalilem2  25516  qsalrel  42223