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Theorem oprssdm 7615
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 5702 . 2 Rel (𝑆 × 𝑆)
2 opelxp 5720 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) ↔ (𝑥𝑆𝑦𝑆))
3 df-ov 7435 . . . . 5 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4 oprssdm.2 . . . . 5 ((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)
53, 4eqeltrrid 2845 . . . 4 ((𝑥𝑆𝑦𝑆) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆)
6 oprssdm.1 . . . . . 6 ¬ ∅ ∈ 𝑆
7 ndmfv 6940 . . . . . . 7 (¬ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹 → (𝐹‘⟨𝑥, 𝑦⟩) = ∅)
87eleq1d 2825 . . . . . 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 5796 1 (𝑆 × 𝑆) ⊆ dom 𝐹
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2107  wss 3950  c0 4332  cop 4631   × cxp 5682  dom cdm 5684  cfv 6560  (class class class)co 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-dm 5694  df-iota 6513  df-fv 6568  df-ov 7435
This theorem is referenced by:  dmaddsr  11126  dmmulsr  11127  axaddf  11186  axmulf  11187
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