Theorem relresOLD 5998

Description: Obsolete version of relres 5997 as of 16-Oct-2026. (Contributed by NM, 2-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
relresOLD	⊢ Rel (𝐴 ↾ 𝐵)

Proof of Theorem relresOLD

Step	Hyp	Ref	Expression
1		df-res 5663	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		inss2 4133	. . 3 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ (𝐵 × V)
3	1, 2	eqsstri 3955	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ (𝐵 × V)
4		relxp 5669	. 2 ⊢ Rel (𝐵 × V)
5		relss 5759	. 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ (𝐵 × V) → (Rel (𝐵 × V) → Rel (𝐴 ↾ 𝐵)))
6	3, 4, 5	mp2 9	1 ⊢ Rel (𝐴 ↾ 𝐵)

This proof depends on syntax axioms:  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088   × cxp 5649   ↾ cres 5653  Rel wrel 5656

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-opab 5167  df-xp 5657  df-rel 5658  df-res 5663

This theorem is used by: (None)