Theorem relssi 5788

Description: Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)

Hypotheses

Ref	Expression
relssi.1	⊢ Rel 𝐴
relssi.2	⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)

Assertion

Ref	Expression
relssi	⊢ 𝐴 ⊆ 𝐵

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

Proof of Theorem relssi

Step	Hyp	Ref	Expression
1		relssi.1	. . 3 ⊢ Rel 𝐴
2		ssrel 5783	. . 3 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
4		relssi.2	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)
5	4	ax-gen 1798	. 2 ⊢ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)
6	3, 5	mpgbir 1802	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1540   ∈ wcel 2107   ⊆ wss 3949  ⟨cop 4635  Rel wrel 5682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-opab 5212  df-xp 5683  df-rel 5684

This theorem is referenced by:  xpsspw  5810  oprssdm  7588  dftpos4  8230  enssdom  8973  idssen  8993  ttrcltr  9711  txuni2  23069  acycgr0v  34139  prclisacycgr  34142  txpss3v  34850  pprodss4v  34856  bj-idres  36041  xrnss3v  37242  aoprssdm  45910