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Theorem relssi 5653
Description: Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)
Hypotheses
Ref Expression
relssi.1 Rel 𝐴
relssi.2 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)
Assertion
Ref Expression
relssi 𝐴𝐵
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem relssi
StepHypRef Expression
1 relssi.1 . . 3 Rel 𝐴
2 ssrel 5650 . . 3 (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
31, 2ax-mp 5 . 2 (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
4 relssi.2 . . 3 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)
54ax-gen 1787 . 2 𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)
63, 5mpgbir 1791 1 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1526  wcel 2105  wss 3933  cop 4563  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-in 3940  df-ss 3949  df-opab 5120  df-xp 5554  df-rel 5555
This theorem is referenced by:  xpsspw  5675  oprssdm  7318  dftpos4  7900  enssdom  8522  idssen  8542  txuni2  22101  acycgr0v  32292  prclisacycgr  32295  txpss3v  33236  pprodss4v  33242  bj-idres  34344  xrnss3v  35504  aoprssdm  43278
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