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Theorem relssi 5368
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 5364 . . 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 1871 . 2 𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)
63, 5mpgbir 1875 1 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1630  wcel 2139  wss 3715  cop 4327  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-in 3722  df-ss 3729  df-opab 4865  df-xp 5272  df-rel 5273
This theorem is referenced by:  xpsspw  5389  oprssdm  6980  resiexg  7267  dftpos4  7540  enssdom  8146  idssen  8166  txuni2  21570  txpss3v  32291  pprodss4v  32297  xrnss3v  34457  aoprssdm  41788
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