Theorem elecreseq 8671

Description: The restricted coset of 𝐵 when 𝐵 is an element of the restriction. (Contributed by Peter Mazsa, 16-Oct-2018.)

Assertion

Ref	Expression
elecreseq	⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅)

Proof of Theorem elecreseq

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elecres 8670	. . . . 5 ⊢ (𝑦 ∈ V → (𝑦 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑦)))
2	1	elv 3441	. . . 4 ⊢ (𝑦 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑦))
3	2	baib 535	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝑦 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ 𝐵𝑅𝑦))
4	3	eqabdv 2864	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = {𝑦 ∣ 𝐵𝑅𝑦})
5		dfec2 8625	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵]𝑅 = {𝑦 ∣ 𝐵𝑅𝑦})
6	4, 5	eqtr4d 2769	1 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅)

Syntax hints:   → wi 4   ↔ 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:  elecex  8672  eccnvepres2  38333  eldmqsres  38335  qsresid  38373  ecuncnvepres  38429  dfblockliftmap2  38484