Theorem eqres 37012

Description: Converting a class constant definition by restriction (like df-ers 37336 or df-parts 37438) 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 5147	. 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(𝑆 ↾ 𝐶)𝐵)
3		brres 5980	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴(𝑆 ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵)))
4	2, 3	bitrid 282	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   class class class wbr 5141   ↾ cres 5671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-xp 5675  df-res 5681

This theorem is referenced by:  brers  37340  brparts  37444