Theorem eqres 38341

Description: Converting a class constant definition by restriction (like df-ers 38664 or df-parts 38766) into a binary relation. (Contributed by Peter Mazsa, 1-Oct-2018.)

Hypothesis

Ref	Expression
eqres.1	⊢ 𝑅 = (𝑆 ↾ 𝐶)

Assertion

Ref	Expression
eqres	⊢ (𝐵 ∈ 𝑉 → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵)))

Proof of Theorem eqres

Step	Hyp	Ref	Expression
1		eqres.1	. . 3 ⊢ 𝑅 = (𝑆 ↾ 𝐶)
2	1	breqi 5149	. 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(𝑆 ↾ 𝐶)𝐵)
3		brres 6004	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴(𝑆 ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵)))
4	2, 3	bitrid 283	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   class class class wbr 5143   ↾ cres 5687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-res 5697

This theorem is referenced by:  brers  38668  brparts  38772