Theorem oprssdm 7615

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 5702	. 2 ⊢ Rel (𝑆 × 𝑆)
2		opelxp 5720	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆))
3		df-ov 7435	. . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4		oprssdm.2	. . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)
5	3, 4	eqeltrrid 2845	. . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆)
6		oprssdm.1	. . . . . 6 ⊢ ¬ ∅ ∈ 𝑆
7		ndmfv 6940	. . . . . . 7 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹 → (𝐹‘⟨𝑥, 𝑦⟩) = ∅)
8	7	eleq1d 2825	. . . . . 6 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹 → ((𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
9	6, 8	mtbiri 327	. . . . 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 217	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
13	1, 12	relssi 5796	1 ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹

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