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Theorem ecss 6602
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 (𝜑𝑅 Er 𝑋)
Assertion
Ref Expression
ecss (𝜑 → [𝐴]𝑅𝑋)

Proof of Theorem ecss
StepHypRef Expression
1 df-ec 6561 . . 3 [𝐴]𝑅 = (𝑅 “ {𝐴})
2 imassrn 4999 . . 3 (𝑅 “ {𝐴}) ⊆ ran 𝑅
31, 2eqsstri 3202 . 2 [𝐴]𝑅 ⊆ ran 𝑅
4 ecss.1 . . 3 (𝜑𝑅 Er 𝑋)
5 errn 6581 . . 3 (𝑅 Er 𝑋 → ran 𝑅 = 𝑋)
64, 5syl 14 . 2 (𝜑 → ran 𝑅 = 𝑋)
73, 6sseqtrid 3220 1 (𝜑 → [𝐴]𝑅𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wss 3144  {csn 3607  ran crn 4645  cima 4647   Er wer 6556  [cec 6557
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 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-er 6559  df-ec 6561
This theorem is referenced by:  qsss  6620  divsfvalg  12805
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