Theorem relssdv 5744

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 1930	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		relssdv.1	. . 3 ⊢ (𝜑 → Rel 𝐴)
4		ssrel 5739	. . 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 1540   ∈ wcel 2114   ⊆ wss 3889  ⟨cop 4573  Rel wrel 5636

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-ss 3906  df-opab 5148  df-xp 5637  df-rel 5638

This theorem is referenced by:  relssres  5987  poirr2  6087  sofld  6151  relssdmrn  6233  funcres2  17865  wunfunc  17868  fthres2  17901  pospo  18309  joindmss  18343  meetdmss  18357  clatl  18474  subrgdvds  20563  opsrtoslem2  22034  txcls  23569  txdis1cn  23600  txkgen  23617  qustgplem  24086  metustid  24519  metustexhalf  24521  ovoliunlem1  25469  dvres2  25879  cvmlift2lem12  35496  dib2dim  41689  dih2dimbALTN  41691  dihmeetlem1N  41736  dihglblem5apreN  41737  dihmeetlem13N  41765  dihjatcclem4  41867