Theorem oprssdm 7584

Description: Domain of closure of an operation. (Contributed by NM, 24-Aug-1995.)

Hypotheses

Ref	Expression
oprssdm.1	⊢ ¬ ∅ ∈ 𝑆
oprssdm.2	⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)

Assertion

Ref	Expression
oprssdm	⊢ (𝑆 × 𝑆) ⊆ dom 𝐹

Distinct variable groups:   𝑥,𝑦,𝑆   𝑥,𝐹,𝑦

Proof of Theorem oprssdm

Step	Hyp	Ref	Expression
1		relxp 5693	. 2 ⊢ Rel (𝑆 × 𝑆)
2		opelxp 5711	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆))
3		df-ov 7408	. . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4		oprssdm.2	. . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)
5	3, 4	eqeltrrid 2838	. . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆)
6		oprssdm.1	. . . . . 6 ⊢ ¬ ∅ ∈ 𝑆
7		ndmfv 6923	. . . . . . 7 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹 → (𝐹‘⟨𝑥, 𝑦⟩) = ∅)
8	7	eleq1d 2818	. . . . . 6 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹 → ((𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
9	6, 8	mtbiri 326	. . . . 5 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹 → ¬ (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆)
10	9	con4i 114	. . . 4 ⊢ ((𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆 → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
11	5, 10	syl 17	. . 3 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
12	2, 11	sylbi 216	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
13	1, 12	relssi 5785	1 ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∈ wcel 2106   ⊆ wss 3947  ∅c0 4321  ⟨cop 4633   × cxp 5673  dom cdm 5675  ‘cfv 6540  (class class class)co 7405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-dm 5685  df-iota 6492  df-fv 6548  df-ov 7408

This theorem is referenced by:  dmaddsr  11076  dmmulsr  11077  axaddf  11136  axmulf  11137