Theorem oprssdm 7581

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 5670	. 2 ⊢ Rel (𝑆 × 𝑆)
2		opelxp 5688	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆))
3		df-ov 7403	. . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4		oprssdm.2	. . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)
5	3, 4	eqeltrrid 2870	. . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆)
6		oprssdm.1	. . . . . 6 ⊢ ¬ ∅ ∈ 𝑆
7		ndmfv 6903	. . . . . . 7 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹 → (𝐹‘⟨𝑥, 𝑦⟩) = ∅)
8	7	eleq1d 2850	. . . . . 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 5764	1 ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∈ wcel 2145   ⊆ wss 3907  ∅c0 4288  ⟨cop 4591   × cxp 5650  dom cdm 5652  ‘cfv 6525  (class class class)co 7400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-dm 5662  df-iota 6481  df-fv 6533  df-ov 7403

This theorem is referenced by:  dmaddsr  11058  dmmulsr  11059  axaddf  11118  axmulf  11119