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Theorem ndmaovcl 46630
Description: The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7613 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 5718 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
3 ndmaov.1 . . . . . 6 dom 𝐹 = (𝑆 × 𝑆)
43eqcomi 2737 . . . . 5 (𝑆 × 𝑆) = dom 𝐹
54eleq2i 2821 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6 ndmaovcl.3 . . . . 5 ((𝐴𝐹𝐵)) ∈ V
7 ndmaov 46610 . . . . 5 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
8 eleq1 2817 . . . . . . 7 ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V ↔ V ∈ V))
98biimpd 228 . . . . . 6 ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V → V ∈ V))
10 vprc 5319 . . . . . . 7 ¬ V ∈ V
1110pm2.21i 119 . . . . . 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 329 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
152, 14sylnbir 330 . 2 (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
161, 15pm2.61i 182 1 ((𝐴𝐹𝐵)) ∈ 𝑆
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3473  cop 4638   × cxp 5680  dom cdm 5682   ((caov 46545
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-iota 6505  df-fun 6555  df-fv 6561  df-aiota 46512  df-dfat 46546  df-afv 46547  df-aov 46548
This theorem is referenced by: (None)
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