Theorem ssrel3 5799

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 5795	. 2 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
2		df-br 5149	. . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3		df-br 5149	. . . 4 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
4	2, 3	imbi12i 350	. . 3 ⊢ ((𝑥𝐴𝑦 → 𝑥𝐵𝑦) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
5	4	2albii 1817	. 2 ⊢ (∀𝑥∀𝑦(𝑥𝐴𝑦 → 𝑥𝐵𝑦) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
6	1, 5	bitr4di 289	1 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → 𝑥𝐵𝑦)))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   ∈ wcel 2106   ⊆ wss 3963  ⟨cop 4637   class class class wbr 5148  Rel wrel 5694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696

This theorem is referenced by:  cotrg  6130  cotrgOLD  6131  cnvsym  6135  cnvsymOLD  6136  eqrel2  38281  inxpss  38293  inxpss2  38297  cnvref5  38333  cocossss  38418