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Theorem relssi 5134
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 5130 . . 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 1713 . 2 𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)
63, 5mpgbir 1717 1 𝐴𝐵
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:  xpsspw  5156  oprssdm  6713  resiexg  6994  dftpos4  7258  enssdom  7866  idssen  7886  txuni2  21178  txpss3v  31155  pprodss4v  31161  aoprssdm  39931
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