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Theorem ndmaovcl 41808
Description: The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 6986 where it is required that the domain contains the empty set (∅ ∈ 𝑆). (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypotheses
Ref Expression
ndmaov.1 dom 𝐹 = (𝑆 × 𝑆)
ndmaovcl.2 ((𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
ndmaovcl.3 ((𝐴𝐹𝐵)) ∈ V
Assertion
Ref Expression
ndmaovcl ((𝐴𝐹𝐵)) ∈ 𝑆

Proof of Theorem ndmaovcl
StepHypRef Expression
1 ndmaovcl.2 . 2 ((𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
2 opelxp 5304 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
3 ndmaov.1 . . . . . 6 dom 𝐹 = (𝑆 × 𝑆)
43eqcomi 2770 . . . . 5 (𝑆 × 𝑆) = dom 𝐹
54eleq2i 2832 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6 ndmaovcl.3 . . . . 5 ((𝐴𝐹𝐵)) ∈ V
7 ndmaov 41788 . . . . 5 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
8 eleq1 2828 . . . . . . 7 ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V ↔ V ∈ V))
98biimpd 219 . . . . . 6 ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V → V ∈ V))
10 vprc 4949 . . . . . . 7 ¬ V ∈ V
1110pm2.21i 116 . . . . . 6 (V ∈ V → ((𝐴𝐹𝐵)) ∈ 𝑆)
129, 11syl6com 37 . . . . 5 ( ((𝐴𝐹𝐵)) ∈ V → ( ((𝐴𝐹𝐵)) = V → ((𝐴𝐹𝐵)) ∈ 𝑆))
136, 7, 12mpsyl 68 . . . 4 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) ∈ 𝑆)
145, 13sylnbi 319 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
152, 14sylnbir 320 . 2 (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
161, 15pm2.61i 176 1 ((𝐴𝐹𝐵)) ∈ 𝑆
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wcel 2140  Vcvv 3341  cop 4328   × cxp 5265  dom cdm 5267   ((caov 41720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-opab 4866  df-xp 5273  df-fv 6058  df-dfat 41721  df-afv 41722  df-aov 41723
This theorem is referenced by: (None)
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