Theorem aoprssdm 47326

Description: Domain of closure of an operation. In contrast to oprssdm 7533, 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 5637	. 2 ⊢ Rel (𝑆 × 𝑆)
2		opelxp 5655	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆))
3		df-aov 47245	. . . . 5 ⊢ ((𝑥𝐹𝑦)) = (𝐹'''⟨𝑥, 𝑦⟩)
4		aoprssdm.1	. . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦)) ∈ 𝑆)
5	3, 4	eqeltrrid 2838	. . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐹'''⟨𝑥, 𝑦⟩) ∈ 𝑆)
6		afvvdm 47265	. . . 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 217	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
9	1, 8	relssi 5731	1 ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113   ⊆ wss 3898  ⟨cop 4581   × cxp 5617  dom cdm 5619  '''cafv 47241   ((caov 47242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-res 5631  df-iota 6442  df-fun 6488  df-fv 6494  df-aiota 47209  df-dfat 47243  df-afv 47244  df-aov 47245

This theorem is referenced by: (None)