Theorem elecreseq 8743

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 8742	. . . . 5 ⊢ (𝑦 ∈ V → (𝑦 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑦)))
2	1	elv 3468	. . . 4 ⊢ (𝑦 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑦))
3	2	baib 544	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝑦 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ 𝐵𝑅𝑦))
4	3	eqabdv 2902	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = {𝑦 ∣ 𝐵𝑅𝑦})
5		dfec2 8696	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵]𝑅 = {𝑦 ∣ 𝐵𝑅𝑦})
6	4, 5	eqtr4d 2807	1 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  Vcvv 3463   class class class wbr 5113   ↾ cres 5664  [cec 8691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ec 8695

This theorem is referenced by:  elecex  8744  eccnvepres2  38829  eldmqsres  38831  qsresid  38869  ecuncnvepres  38933  dfblockliftmap2  38999