Theorem elecres 35533

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 5881	. . 3 ⊢ Rel (𝑅 ↾ 𝐴)
2		relelec 8333	. . 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 5859	. 2 ⊢ (𝐶 ∈ 𝑉 → (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)))
5	3, 4	syl5bb 285	1 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2110   class class class wbr 5065   ↾ cres 5556  Rel wrel 5559  [cec 8286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-xp 5560  df-rel 5561  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ec 8290

This theorem is referenced by:  ecres  35534  ecres2  35535