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Theorem ndmaovcl 46483
Description: The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7589 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 5705 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
3 ndmaov.1 . . . . . 6 dom 𝐹 = (𝑆 × 𝑆)
43eqcomi 2735 . . . . 5 (𝑆 × 𝑆) = dom 𝐹
54eleq2i 2819 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6 ndmaovcl.3 . . . . 5 ((𝐴𝐹𝐵)) ∈ V
7 ndmaov 46463 . . . . 5 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
8 eleq1 2815 . . . . . . 7 ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V ↔ V ∈ V))
98biimpd 228 . . . . . 6 ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V → V ∈ V))
10 vprc 5308 . . . . . . 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 330 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
152, 14sylnbir 331 . 2 (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
161, 15pm2.61i 182 1 ((𝐴𝐹𝐵)) ∈ 𝑆
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  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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