Theorem ecres 38321

Description: Restricted coset of 𝐵. (Contributed by Peter Mazsa, 9-Dec-2018.)

Assertion

Ref	Expression
ecres	⊢ [𝐵](𝑅 ↾ 𝐴) = {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑥)}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Proof of Theorem ecres

Step	Hyp	Ref	Expression
1		elecres 8670	. . 3 ⊢ (𝑥 ∈ V → (𝑥 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑥)))
2	1	elv 3441	. 2 ⊢ (𝑥 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑥))
3	2	eqabi 2866	1 ⊢ [𝐵](𝑅 ↾ 𝐴) = {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑥)}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  Vcvv 3436   class class class wbr 5089   ↾ cres 5616  [cec 8620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ec 8624

This theorem is referenced by:  eccnvepres  38322