Theorem relssi 5785

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 5780	. . 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 1797	. 2 ⊢ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)
6	3, 5	mpgbir 1801	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539   ∈ wcel 2106   ⊆ wss 3947  ⟨cop 4633  Rel wrel 5680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3954  df-ss 3964  df-opab 5210  df-xp 5681  df-rel 5682

This theorem is referenced by:  xpsspw  5807  oprssdm  7584  dftpos4  8226  enssdom  8969  idssen  8989  ttrcltr  9707  txuni2  23060  acycgr0v  34127  prclisacycgr  34130  txpss3v  34838  pprodss4v  34844  bj-idres  36029  xrnss3v  37230  aoprssdm  45896