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Theorem oprssdm 7614
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 5707 . 2 Rel (𝑆 × 𝑆)
2 opelxp 5725 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) ↔ (𝑥𝑆𝑦𝑆))
3 df-ov 7434 . . . . 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 6942 . . . . . . 7 (¬ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹 → (𝐹‘⟨𝑥, 𝑦⟩) = ∅)
87eleq1d 2824 . . . . . 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 217 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
131, 12relssi 5800 1 (𝑆 × 𝑆) ⊆ dom 𝐹
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2106  wss 3963  c0 4339  cop 4637   × cxp 5687  dom cdm 5689  cfv 6563  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by:  dmaddsr  11123  dmmulsr  11124  axaddf  11183  axmulf  11184
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