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Theorem ecss 6573
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 6534 . . 3 [𝐴]𝑅 = (𝑅 “ {𝐴})
2 imassrn 4980 . . 3 (𝑅 “ {𝐴}) ⊆ ran 𝑅
31, 2eqsstri 3187 . 2 [𝐴]𝑅 ⊆ ran 𝑅
4 ecss.1 . . 3 (𝜑𝑅 Er 𝑋)
5 errn 6554 . . 3 (𝑅 Er 𝑋 → ran 𝑅 = 𝑋)
64, 5syl 14 . 2 (𝜑 → ran 𝑅 = 𝑋)
73, 6sseqtrid 3205 1 (𝜑 → [𝐴]𝑅𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wss 3129  {csn 3592  ran crn 4626  cima 4628   Er wer 6529  [cec 6530
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-xp 4631  df-rel 4632  df-cnv 4633  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-er 6532  df-ec 6534
This theorem is referenced by:  qsss  6591
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