Theorem dfec2 6772

Description: Alternate definition of 𝑅-coset of 𝐴. Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
dfec2	⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑦 ∣ 𝐴𝑅𝑦})

Distinct variable groups:   𝑦,𝐴   𝑦,𝑅

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem dfec2

Step	Hyp	Ref	Expression
1		df-ec 6771	. 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
2		imasng 5129	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
3	1, 2	eqtrid 2279	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑦 ∣ 𝐴𝑅𝑦})

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2205  {cab 2220  {csn 3691   class class class wbr 4111   “ cima 4754  [cec 6767

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-cnv 4759  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-ec 6771

This theorem is referenced by:  eqglact  13963