Theorem elecreseq 8684

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 8683	. . . . 5 ⊢ (𝑦 ∈ V → (𝑦 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑦)))
2	1	elv 3445	. . . 4 ⊢ (𝑦 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑦))
3	2	baib 535	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝑦 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ 𝐵𝑅𝑦))
4	3	eqabdv 2869	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = {𝑦 ∣ 𝐵𝑅𝑦})
5		dfec2 8638	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵]𝑅 = {𝑦 ∣ 𝐵𝑅𝑦})
6	4, 5	eqtr4d 2774	1 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2714  Vcvv 3440   class class class wbr 5098   ↾ cres 5626  [cec 8633

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 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8637

This theorem is referenced by:  elecex  8685  eccnvepres2  38484  eldmqsres  38486  qsresid  38524  ecuncnvepres  38580  dfblockliftmap2  38635