Theorem aoprssdm 39931

Description: Domain of closure of an operation. In contrast to oprssdm 6713, no additional property for S (¬ ∅ ∈ 𝑆) is required! (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypothesis

Ref	Expression
aoprssdm.1	⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦)) ∈ 𝑆)

Assertion

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

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

Proof of Theorem aoprssdm

Step	Hyp	Ref	Expression
1		relxp 5150	. 2 ⊢ Rel (𝑆 × 𝑆)
2		opelxp 5070	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆))
3		df-aov 39847	. . . . 5 ⊢ ((𝑥𝐹𝑦)) = (𝐹'''⟨𝑥, 𝑦⟩)
4		aoprssdm.1	. . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦)) ∈ 𝑆)
5	3, 4	syl5eqelr 2693	. . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐹'''⟨𝑥, 𝑦⟩) ∈ 𝑆)
6		afvvdm 39870	. . . 4 ⊢ ((𝐹'''⟨𝑥, 𝑦⟩) ∈ 𝑆 → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
7	5, 6	syl 17	. . 3 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
8	2, 7	sylbi 206	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
9	1, 8	relssi 5134	1 ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977   ⊆ wss 3540  ⟨cop 4131   × cxp 5036  dom cdm 5038  '''cafv 39843   ((caov 39844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-rel 5045  df-fv 5812  df-dfat 39845  df-afv 39846  df-aov 39847

This theorem is referenced by: (None)