Theorem resss2 38180

Description: The restriction of a class to a class is included in the Cartesian product of the restricting class with the universal class. (Contributed by BJ, 16-Oct-2026.)

Assertion

Ref	Expression
resss2	⊢ (𝑅 ↾ 𝐴) ⊆ (𝐴 × V)

Proof of Theorem resss2

Step	Hyp	Ref	Expression
1		df-res 5670	. 2 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V))
2		inss2 4137	. 2 ⊢ (𝑅 ∩ (𝐴 × V)) ⊆ (𝐴 × V)
3	1, 2	eqsstri 3959	1 ⊢ (𝑅 ↾ 𝐴) ⊆ (𝐴 × V)

This proof depends on syntax axioms:  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092   × cxp 5656   ↾ cres 5660

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-res 5670

This theorem is used by:  bj-brresdm  38181