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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ndmaovcl | Structured version Visualization version GIF version |
Description: The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7544 where it is required that the domain contains the empty set (∅ ∈ 𝑆). (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
ndmaov.1 | ⊢ dom 𝐹 = (𝑆 × 𝑆) |
ndmaovcl.2 | ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆) |
ndmaovcl.3 | ⊢ ((𝐴𝐹𝐵)) ∈ V |
Ref | Expression |
---|---|
ndmaovcl | ⊢ ((𝐴𝐹𝐵)) ∈ 𝑆 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ndmaovcl.2 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆) | |
2 | opelxp 5674 | . . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) | |
3 | ndmaov.1 | . . . . . 6 ⊢ dom 𝐹 = (𝑆 × 𝑆) | |
4 | 3 | eqcomi 2746 | . . . . 5 ⊢ (𝑆 × 𝑆) = dom 𝐹 |
5 | 4 | eleq2i 2830 | . . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) |
6 | ndmaovcl.3 | . . . . 5 ⊢ ((𝐴𝐹𝐵)) ∈ V | |
7 | ndmaov 45489 | . . . . 5 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V) | |
8 | eleq1 2826 | . . . . . . 7 ⊢ ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V ↔ V ∈ V)) | |
9 | 8 | biimpd 228 | . . . . . 6 ⊢ ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V → V ∈ V)) |
10 | vprc 5277 | . . . . . . 7 ⊢ ¬ V ∈ V | |
11 | 10 | pm2.21i 119 | . . . . . 6 ⊢ (V ∈ V → ((𝐴𝐹𝐵)) ∈ 𝑆) |
12 | 9, 11 | syl6com 37 | . . . . 5 ⊢ ( ((𝐴𝐹𝐵)) ∈ V → ( ((𝐴𝐹𝐵)) = V → ((𝐴𝐹𝐵)) ∈ 𝑆)) |
13 | 6, 7, 12 | mpsyl 68 | . . . 4 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) ∈ 𝑆) |
14 | 5, 13 | sylnbi 330 | . . 3 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆) |
15 | 2, 14 | sylnbir 331 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆) |
16 | 1, 15 | pm2.61i 182 | 1 ⊢ ((𝐴𝐹𝐵)) ∈ 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3448 ⟨cop 4597 × cxp 5636 dom cdm 5638 ((caov 45424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-res 5650 df-iota 6453 df-fun 6503 df-fv 6509 df-aiota 45391 df-dfat 45425 df-afv 45426 df-aov 45427 |
This theorem is referenced by: (None) |
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