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Theorem ndmaovcl 40603
 Description: The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 6775 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 5108 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
3 ndmaov.1 . . . . . 6 dom 𝐹 = (𝑆 × 𝑆)
43eqcomi 2630 . . . . 5 (𝑆 × 𝑆) = dom 𝐹
54eleq2i 2690 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6 ndmaovcl.3 . . . . 5 ((𝐴𝐹𝐵)) ∈ V
7 ndmaov 40583 . . . . 5 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
8 eleq1 2686 . . . . . . 7 ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V ↔ V ∈ V))
98biimpd 219 . . . . . 6 ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V → V ∈ V))
10 vprc 4758 . . . . . . 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 320 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
152, 14sylnbir 321 . 2 (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
161, 15pm2.61i 176 1 ((𝐴𝐹𝐵)) ∈ 𝑆
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186  ⟨cop 4156   × cxp 5074  dom cdm 5076   ((caov 40515 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-opab 4676  df-xp 5082  df-fv 5857  df-dfat 40516  df-afv 40517  df-aov 40518 This theorem is referenced by: (None)
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