Theorem relresOLD 6013

Description: Obsolete version of relres 6012 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 5670	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		inss2 4137	. . 3 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ (𝐵 × V)
3	1, 2	eqsstri 3959	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ (𝐵 × V)
4		relxp 5676	. 2 ⊢ Rel (𝐵 × V)
5		relss 5766	. 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 3454   ⊆ wss 3900   ∩ cin 4092   × cxp 5656   ↾ cres 5660  Rel wrel 5663

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-opab 5174  df-xp 5664  df-rel 5665  df-res 5670

This theorem is used by: (None)