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Theorem aoprssdm 45508
Description: Domain of closure of an operation. In contrast to oprssdm 7540, 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 5656 . 2 Rel (𝑆 × 𝑆)
2 opelxp 5674 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) ↔ (𝑥𝑆𝑦𝑆))
3 df-aov 45427 . . . . 5 ((𝑥𝐹𝑦)) = (𝐹'''⟨𝑥, 𝑦⟩)
4 aoprssdm.1 . . . . 5 ((𝑥𝑆𝑦𝑆) → ((𝑥𝐹𝑦)) ∈ 𝑆)
53, 4eqeltrrid 2843 . . . 4 ((𝑥𝑆𝑦𝑆) → (𝐹'''⟨𝑥, 𝑦⟩) ∈ 𝑆)
6 afvvdm 45447 . . . 4 ((𝐹'''⟨𝑥, 𝑦⟩) ∈ 𝑆 → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
75, 6syl 17 . . 3 ((𝑥𝑆𝑦𝑆) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
82, 7sylbi 216 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
91, 8relssi 5748 1 (𝑆 × 𝑆) ⊆ dom 𝐹
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wss 3915  cop 4597   × cxp 5636  dom cdm 5638  '''cafv 45423   ((caov 45424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-res 5650  df-iota 6453  df-fun 6503  df-fv 6509  df-aiota 45391  df-dfat 45425  df-afv 45426  df-aov 45427
This theorem is referenced by: (None)
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