Theorem ecres 38255

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 8680	. . 3 ⊢ (𝑥 ∈ V → (𝑥 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑥)))
2	1	elv 3443	. 2 ⊢ (𝑥 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑥))
3	2	eqabi 2863	1 ⊢ [𝐵](𝑅 ↾ 𝐴) = {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑥)}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  Vcvv 3438   class class class wbr 5095   ↾ cres 5625  [cec 8630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ec 8634

This theorem is referenced by:  eccnvepres  38256