Theorem eqres 39049

Description: Converting a class constant definition by restriction (like df-ers 39457 or df-parts 39577) 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 5117	. 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(𝑆 ↾ 𝐶)𝐵)
3		brres 5987	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴(𝑆 ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵)))
4	2, 3	bitrid 286	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   class class class wbr 5111   ↾ cres 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-res 5675

This theorem is used by:  brers  39461  brparts  39583