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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 → (𝐴𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem ssrel3
StepHypRef Expression
1 ssrel 5791 . 2 (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
2 df-br 5143 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3 df-br 5143 . . . 4 (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
42, 3imbi12i 350 . . 3 ((𝑥𝐴𝑦𝑥𝐵𝑦) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
542albii 1819 . 2 (∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
61, 5bitr4di 289 1 (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦)))
Colors of variables: wff setvar class
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
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