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Theorem aoprssdm 41784
Description: Domain of closure of an operation. In contrast to oprssdm 6976, no additional property for S (¬ ∅ ∈ 𝑆) is required! (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypothesis
Ref Expression
aoprssdm.1 ((𝑥𝑆𝑦𝑆) → ((𝑥𝐹𝑦)) ∈ 𝑆)
Assertion
Ref Expression
aoprssdm (𝑆 × 𝑆) ⊆ dom 𝐹
Distinct variable groups:   𝑥,𝑦,𝑆   𝑥,𝐹,𝑦

Proof of Theorem aoprssdm
StepHypRef Expression
1 relxp 5279 . 2 Rel (𝑆 × 𝑆)
2 opelxp 5299 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) ↔ (𝑥𝑆𝑦𝑆))
3 df-aov 41700 . . . . 5 ((𝑥𝐹𝑦)) = (𝐹'''⟨𝑥, 𝑦⟩)
4 aoprssdm.1 . . . . 5 ((𝑥𝑆𝑦𝑆) → ((𝑥𝐹𝑦)) ∈ 𝑆)
53, 4syl5eqelr 2840 . . . 4 ((𝑥𝑆𝑦𝑆) → (𝐹'''⟨𝑥, 𝑦⟩) ∈ 𝑆)
6 afvvdm 41723 . . . 4 ((𝐹'''⟨𝑥, 𝑦⟩) ∈ 𝑆 → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
75, 6syl 17 . . 3 ((𝑥𝑆𝑦𝑆) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
82, 7sylbi 207 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
91, 8relssi 5364 1 (𝑆 × 𝑆) ⊆ dom 𝐹
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2135  wss 3711  cop 4323   × cxp 5260  dom cdm 5262  '''cafv 41696   ((caov 41697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pr 5051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-opab 4861  df-xp 5268  df-rel 5269  df-fv 6053  df-dfat 41698  df-afv 41699  df-aov 41700
This theorem is referenced by: (None)
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