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Theorem ndmaovrcl 46484
Description: Reverse closure law, in contrast to ndmovrcl 7590 where it is required that the operation's domain doesn't contain the empty set (¬ ∅ ∈ 𝑆), no additional asumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypothesis
Ref Expression
ndmaov.1 dom 𝐹 = (𝑆 × 𝑆)
Assertion
Ref Expression
ndmaovrcl ( ((𝐴𝐹𝐵)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))

Proof of Theorem ndmaovrcl
StepHypRef Expression
1 aovvdm 46465 . 2 ( ((𝐴𝐹𝐵)) ∈ 𝑆 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
2 opelxp 5705 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
32biimpi 215 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) → (𝐴𝑆𝐵𝑆))
4 ndmaov.1 . . 3 dom 𝐹 = (𝑆 × 𝑆)
53, 4eleq2s 2845 . 2 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆))
61, 5syl 17 1 ( ((𝐴𝐹𝐵)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cop 4629   × cxp 5667  dom cdm 5669   ((caov 46398
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 6489  df-fun 6539  df-fv 6545  df-aiota 46365  df-dfat 46399  df-afv 46400  df-aov 46401
This theorem is referenced by: (None)
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