Theorem bj-resss2 38057

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
bj-resss2	⊢ (𝑅 ↾ 𝐴) ⊆ (𝐴 × V)

Proof of Theorem bj-resss2

Step	Hyp	Ref	Expression
1		df-res 5662	. 2 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V))
2		inss2 4133	. 2 ⊢ (𝑅 ∩ (𝐴 × V)) ⊆ (𝐴 × V)
3	1, 2	eqsstri 3955	1 ⊢ (𝑅 ↾ 𝐴) ⊆ (𝐴 × V)

This proof depends on syntax axioms:  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088   × cxp 5648   ↾ cres 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-in 4090  df-res 5662

This theorem is used by: (None)