Theorem ssrel3 5736

Description: Subclass relation in another form when the subclass is a relation. (Contributed by Peter Mazsa, 16-Feb-2019.)

Assertion

Ref	Expression
ssrel3	⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → 𝑥𝐵𝑦)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem ssrel3

Step	Hyp	Ref	Expression
1		ssrel 5733	. 2 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
2		df-br 5080	. . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3		df-br 5080	. . . 4 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
4	2, 3	imbi12i 351	. . 3 ⊢ ((𝑥𝐴𝑦 → 𝑥𝐵𝑦) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
5	4	2albii 1827	. 2 ⊢ (∀𝑥∀𝑦(𝑥𝐴𝑦 → 𝑥𝐵𝑦) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
6	1, 5	bitr4di 290	1 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → 𝑥𝐵𝑦)))

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1545   ∈ wcel 2119   ⊆ wss 3890  ⟨cop 4568   class class class wbr 5079  Rel wrel 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-ss 3907  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632

This theorem is referenced by:  cotrg  6068  cnvsym  6071  eqrel2  38679  inxpss  38691  inxpss2  38695  cnvref5  38725  cocossss  38900