Theorem relssdv 5731

Description: Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)

Hypotheses

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

Assertion

Ref	Expression
relssdv	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

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

Proof of Theorem relssdv

Step	Hyp	Ref	Expression
1		relssdv.2	. . 3 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
2	1	alrimivv 1928	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		relssdv.1	. . 3 ⊢ (𝜑 → Rel 𝐴)
4		ssrel 5726	. . 3 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
5	3, 4	syl 17	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
6	2, 5	mpbird 257	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   ∈ wcel 2109   ⊆ wss 3903  ⟨cop 4583  Rel wrel 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3438  df-ss 3920  df-opab 5155  df-xp 5625  df-rel 5626

This theorem is referenced by:  relssres  5973  poirr2  6073  sofld  6136  relssdmrn  6217  funcres2  17805  wunfunc  17808  fthres2  17841  pospo  18249  joindmss  18283  meetdmss  18297  clatl  18414  subrgdvds  20471  opsrtoslem2  21961  txcls  23489  txdis1cn  23520  txkgen  23537  qustgplem  24006  metustid  24440  metustexhalf  24442  ovoliunlem1  25401  dvres2  25811  cvmlift2lem12  35307  dib2dim  41242  dih2dimbALTN  41244  dihmeetlem1N  41289  dihglblem5apreN  41290  dihmeetlem13N  41318  dihjatcclem4  41420