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Theorem relssdv 5801
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 (𝜑𝐴𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦

Proof of Theorem relssdv
StepHypRef Expression
1 relssdv.2 . . 3 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
21alrimivv 1926 . 2 (𝜑 → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3 relssdv.1 . . 3 (𝜑 → Rel 𝐴)
4 ssrel 5795 . . 3 (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
53, 4syl 17 . 2 (𝜑 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
62, 5mpbird 257 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  wcel 2106  wss 3963  cop 4637  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-opab 5211  df-xp 5695  df-rel 5696
This theorem is referenced by:  relssres  6042  poirr2  6147  sofld  6209  relssdmrn  6290  relssdmrnOLD  6291  funcres2  17949  wunfunc  17952  wunfuncOLD  17953  fthres2  17986  pospo  18403  joindmss  18437  meetdmss  18451  clatl  18566  subrgdvds  20603  opsrtoslem2  22098  txcls  23628  txdis1cn  23659  txkgen  23676  qustgplem  24145  metustid  24583  metustexhalf  24585  ovoliunlem1  25551  dvres2  25962  cvmlift2lem12  35299  dib2dim  41226  dih2dimbALTN  41228  dihmeetlem1N  41273  dihglblem5apreN  41274  dihmeetlem13N  41302  dihjatcclem4  41404
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