Theorem brresi 33838

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 5561	. 2 ⊢ (𝑅 ↾ 𝐶) ⊆ 𝑅
2	1	ssbri 4831	1 ⊢ (𝐴(𝑅 ↾ 𝐶)𝐵 → 𝐴𝑅𝐵)

Syntax hints:   → wi 4   ∈ wcel 2145  Vcvv 3351   class class class wbr 4786   ↾ cres 5251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-ss 3737  df-br 4787  df-res 5261

This theorem is referenced by: (None)