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Theorem relssdv 5246
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 1896 . 2 (𝜑 → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3 relssdv.1 . . 3 (𝜑 → Rel 𝐴)
4 ssrel 5241 . . 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 247 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  wcel 2030  wss 3607  cop 4216  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-in 3614  df-ss 3621  df-opab 4746  df-xp 5149  df-rel 5150
This theorem is referenced by:  relssres  5472  poirr2  5555  sofld  5616  relssdmrn  5694  funcres2  16605  wunfunc  16606  fthres2  16639  pospo  17020  joindmss  17054  meetdmss  17068  clatl  17163  subrgdvds  18842  opsrtoslem2  19533  txcls  21455  txdis1cn  21486  txkgen  21503  qustgplem  21971  metustid  22406  metustexhalf  22408  ovoliunlem1  23316  dvres2  23721  cvmlift2lem12  31422  dib2dim  36849  dih2dimbALTN  36851  dihmeetlem1N  36896  dihglblem5apreN  36897  dihmeetlem13N  36925  dihjatcclem4  37027
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