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Theorem ecss 6550
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 6511 . . 3 [𝐴]𝑅 = (𝑅 “ {𝐴})
2 imassrn 4962 . . 3 (𝑅 “ {𝐴}) ⊆ ran 𝑅
31, 2eqsstri 3179 . 2 [𝐴]𝑅 ⊆ ran 𝑅
4 ecss.1 . . 3 (𝜑𝑅 Er 𝑋)
5 errn 6531 . . 3 (𝑅 Er 𝑋 → ran 𝑅 = 𝑋)
64, 5syl 14 . 2 (𝜑 → ran 𝑅 = 𝑋)
73, 6sseqtrid 3197 1 (𝜑 → [𝐴]𝑅𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wss 3121  {csn 3581  ran crn 4610  cima 4612   Er wer 6506  [cec 6507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-cnv 4617  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-er 6509  df-ec 6511
This theorem is referenced by:  qsss  6568
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