Theorem elecres 37656

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 6003	. . 3 ⊢ Rel (𝑅 ↾ 𝐴)
2		relelec 8747	. . 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 5981	. 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 205   ∧ wa 395   ∈ wcel 2098   class class class wbr 5141   ↾ cres 5671  Rel wrel 5674  [cec 8700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8704

This theorem is referenced by:  ecres  37657  ecres2  37658