Theorem oprssdm 7599

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 5677	. 2 ⊢ Rel (𝑆 × 𝑆)
2		opelxp 5695	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆))
3		df-ov 7420	. . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4		oprssdm.2	. . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)
5	3, 4	eqeltrrid 2867	. . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆)
6		oprssdm.1	. . . . . 6 ⊢ ¬ ∅ ∈ 𝑆
7		ndmfv 6914	. . . . . . 7 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹 → (𝐹‘⟨𝑥, 𝑦⟩) = ∅)
8	7	eleq1d 2847	. . . . . 6 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹 → ((𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
9	6, 8	mtbiri 330	. . . . 5 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹 → ¬ (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆)
10	9	con4i 115	. . . 4 ⊢ ((𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆 → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
11	5, 10	syl 18	. . 3 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
12	2, 11	sylbi 220	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
13	1, 12	relssi 5771	1 ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∈ wcel 2145   ⊆ wss 3902  ∅c0 4282  ⟨cop 4593   × cxp 5657  dom cdm 5659  ‘cfv 6537  (class class class)co 7417

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-dm 5669  df-iota 6493  df-fv 6545  df-ov 7420

This theorem is used by:  dmaddsr  11098  dmmulsr  11099  axaddf  11158  axmulf  11159