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Theorem relssdv 5135
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 1843 . 2 (𝜑 → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3 relssdv.1 . . 3 (𝜑 → Rel 𝐴)
4 ssrel 5130 . . 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 246 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wal 1473  wcel 1977  wss 3540  cop 4131  Rel wrel 5043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554  df-opab 4644  df-xp 5044  df-rel 5045
This theorem is referenced by:  relssres  5357  poirr2  5439  sofld  5500  relssdmrn  5573  funcres2  16381  wunfunc  16382  fthres2  16415  pospo  16796  joindmss  16830  meetdmss  16844  clatl  16939  subrgdvds  18617  opsrtoslem2  19306  txcls  21217  txdis1cn  21248  txkgen  21265  qustgplem  21734  metustid  22169  metustexhalf  22171  ovoliunlem1  23077  dvres2  23482  cvmlift2lem12  30550  dib2dim  35550  dih2dimbALTN  35552  dihmeetlem1N  35597  dihglblem5apreN  35598  dihmeetlem13N  35626  dihjatcclem4  35728
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