Theorem ecss 6630

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 6589	. . 3 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
2		imassrn 5016	. . 3 ⊢ (𝑅 “ {𝐴}) ⊆ ran 𝑅
3	1, 2	eqsstri 3211	. 2 ⊢ [𝐴]𝑅 ⊆ ran 𝑅
4		ecss.1	. . 3 ⊢ (𝜑 → 𝑅 Er 𝑋)
5		errn 6609	. . 3 ⊢ (𝑅 Er 𝑋 → ran 𝑅 = 𝑋)
6	4, 5	syl 14	. 2 ⊢ (𝜑 → ran 𝑅 = 𝑋)
7	3, 6	sseqtrid 3229	1 ⊢ (𝜑 → [𝐴]𝑅 ⊆ 𝑋)

Syntax hints:   → wi 4   = wceq 1364   ⊆ wss 3153  {csn 3618  ran crn 4660   “ cima 4662   Er wer 6584  [cec 6585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-er 6587  df-ec 6589

This theorem is referenced by:  qsss  6648  divsfval  12911  divsfvalg  12912