Theorem brresi 33820

Description: Restriction of a binary relation. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypothesis

Ref	Expression
brresi.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
brresi	⊢ (𝐴(𝑅 ↾ 𝐶)𝐵 → 𝐴𝑅𝐵)

Proof of Theorem brresi

Step	Hyp	Ref	Expression
1		resss 5625	. 2 ⊢ (𝑅 ↾ 𝐶) ⊆ 𝑅
2	1	ssbri 4889	1 ⊢ (𝐴(𝑅 ↾ 𝐶)𝐵 → 𝐴𝑅𝐵)

Syntax hints:   → wi 4   ∈ wcel 2156  Vcvv 3391   class class class wbr 4844   ↾ cres 5313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-v 3393  df-in 3776  df-ss 3783  df-br 4845  df-res 5323

This theorem is referenced by: (None)