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Theorem ndmaovrcl 44229
Description: Reverse closure law, in contrast to ndmovrcl 7350 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 44210 . 2 ( ((𝐴𝐹𝐵)) ∈ 𝑆 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
2 opelxp 5561 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
32biimpi 219 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) → (𝐴𝑆𝐵𝑆))
4 ndmaov.1 . . 3 dom 𝐹 = (𝑆 × 𝑆)
53, 4eleq2s 2851 . 2 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆))
61, 5syl 17 1 ( ((𝐴𝐹𝐵)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  cop 4522   × cxp 5523  dom cdm 5525   ((caov 44143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-int 4837  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-res 5537  df-iota 6297  df-fun 6341  df-fv 6347  df-aiota 44109  df-dfat 44144  df-afv 44145  df-aov 44146
This theorem is referenced by: (None)
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