Theorem ssrel3 5795

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 5791	. 2 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
2		df-br 5143	. . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3		df-br 5143	. . . 4 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
4	2, 3	imbi12i 350	. . 3 ⊢ ((𝑥𝐴𝑦 → 𝑥𝐵𝑦) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
5	4	2albii 1819	. 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 1537   ∈ wcel 2107   ⊆ wss 3950  ⟨cop 4631   class class class wbr 5142  Rel wrel 5689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-ss 3967  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691

This theorem is referenced by:  cotrg  6126  cotrgOLD  6127  cnvsym  6131  cnvsymOLD  6132  eqrel2  38301  inxpss  38313  inxpss2  38317  cnvref5  38353  cocossss  38438