Theorem elecres 8681

Description: Elementhood in the restricted coset of 𝐵. (Contributed by Peter Mazsa, 21-Sep-2018.)

Assertion

Ref	Expression
elecres	⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)))

Proof of Theorem elecres

Step	Hyp	Ref	Expression
1		relres 5962	. . 3 ⊢ Rel (𝑅 ↾ 𝐴)
2		relelec 8680	. . 3 ⊢ (Rel (𝑅 ↾ 𝐴) → (𝐶 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ 𝐵(𝑅 ↾ 𝐴)𝐶))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐶 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ 𝐵(𝑅 ↾ 𝐴)𝐶)
4		brres 5943	. 2 ⊢ (𝐶 ∈ 𝑉 → (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)))
5	3, 4	bitrid 283	1 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2113   class class class wbr 5096   ↾ cres 5624  Rel wrel 5627  [cec 8631

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 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

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 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ec 8635

This theorem is referenced by:  elecreseq  8682  ecres  38417