MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oprssdm Structured version   Visualization version   GIF version

Theorem oprssdm 7444
Description: Domain of closure of an operation. (Contributed by NM, 24-Aug-1995.)
Hypotheses
Ref Expression
oprssdm.1 ¬ ∅ ∈ 𝑆
oprssdm.2 ((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)
Assertion
Ref Expression
oprssdm (𝑆 × 𝑆) ⊆ dom 𝐹
Distinct variable groups:   𝑥,𝑦,𝑆   𝑥,𝐹,𝑦

Proof of Theorem oprssdm
StepHypRef Expression
1 relxp 5603 . 2 Rel (𝑆 × 𝑆)
2 opelxp 5621 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) ↔ (𝑥𝑆𝑦𝑆))
3 df-ov 7271 . . . . 5 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4 oprssdm.2 . . . . 5 ((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)
53, 4eqeltrrid 2844 . . . 4 ((𝑥𝑆𝑦𝑆) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆)
6 oprssdm.1 . . . . . 6 ¬ ∅ ∈ 𝑆
7 ndmfv 6797 . . . . . . 7 (¬ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹 → (𝐹‘⟨𝑥, 𝑦⟩) = ∅)
87eleq1d 2823 . . . . . 6 (¬ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹 → ((𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
96, 8mtbiri 327 . . . . 5 (¬ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹 → ¬ (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆)
109con4i 114 . . . 4 ((𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆 → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
115, 10syl 17 . . 3 ((𝑥𝑆𝑦𝑆) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
122, 11sylbi 216 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
131, 12relssi 5691 1 (𝑆 × 𝑆) ⊆ dom 𝐹
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wcel 2106  wss 3887  c0 4257  cop 4568   × cxp 5583  dom cdm 5585  cfv 6427  (class class class)co 7268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pr 5351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5075  df-opab 5137  df-xp 5591  df-rel 5592  df-dm 5595  df-iota 6385  df-fv 6435  df-ov 7271
This theorem is referenced by:  dmaddsr  10829  dmmulsr  10830  axaddf  10889  axmulf  10890
  Copyright terms: Public domain W3C validator