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Theorem aoprssdm 46464
Description: Domain of closure of an operation. In contrast to oprssdm 7584, 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 5687 . 2 Rel (𝑆 × 𝑆)
2 opelxp 5705 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) ↔ (𝑥𝑆𝑦𝑆))
3 df-aov 46383 . . . . 5 ((𝑥𝐹𝑦)) = (𝐹'''⟨𝑥, 𝑦⟩)
4 aoprssdm.1 . . . . 5 ((𝑥𝑆𝑦𝑆) → ((𝑥𝐹𝑦)) ∈ 𝑆)
53, 4eqeltrrid 2832 . . . 4 ((𝑥𝑆𝑦𝑆) → (𝐹'''⟨𝑥, 𝑦⟩) ∈ 𝑆)
6 afvvdm 46403 . . . 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 5780 1 (𝑆 × 𝑆) ⊆ dom 𝐹
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  wss 3943  cop 4629   × cxp 5667  dom cdm 5669  '''cafv 46379   ((caov 46380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6488  df-fun 6538  df-fv 6544  df-aiota 46347  df-dfat 46381  df-afv 46382  df-aov 46383
This theorem is referenced by: (None)
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