Theorem elecres 38235

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 6037	. . 3 ⊢ Rel (𝑅 ↾ 𝐴)
2		relelec 8812	. . 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 6018	. 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 2108   class class class wbr 5166   ↾ cres 5702  Rel wrel 5705  [cec 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8767

This theorem is referenced by:  ecres  38236  ecres2  38237