Theorem ecres2 36402

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

Assertion

Ref	Expression
ecres2	⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅)

Proof of Theorem ecres2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elecres 36400	. . . . 5 ⊢ (𝑦 ∈ V → (𝑦 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑦)))
2	1	elv 3437	. . . 4 ⊢ (𝑦 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑦))
3	2	baib 536	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝑦 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ 𝐵𝑅𝑦))
4	3	abbi2dv 2879	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = {𝑦 ∣ 𝐵𝑅𝑦})
5		dfec2 8476	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵]𝑅 = {𝑦 ∣ 𝐵𝑅𝑦})
6	4, 5	eqtr4d 2783	1 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110  {cab 2717  Vcvv 3431   class class class wbr 5079   ↾ cres 5591  [cec 8471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ec 8475

This theorem is referenced by:  eccnvepres2  36407  eldmqsres  36409  qsresid  36448  ecex2  36451