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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 7148 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 5440 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) | |
3 | ndmaov.1 | . . . . . 6 ⊢ dom 𝐹 = (𝑆 × 𝑆) | |
4 | 3 | eqcomi 2782 | . . . . 5 ⊢ (𝑆 × 𝑆) = dom 𝐹 |
5 | 4 | eleq2i 2852 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑆 × 𝑆) ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
6 | ndmaovcl.3 | . . . . 5 ⊢ ((𝐴𝐹𝐵)) ∈ V | |
7 | ndmaov 42818 | . . . . 5 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V) | |
8 | eleq1 2848 | . . . . . . 7 ⊢ ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V ↔ V ∈ V)) | |
9 | 8 | biimpd 221 | . . . . . 6 ⊢ ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V → V ∈ V)) |
10 | vprc 5073 | . . . . . . 7 ⊢ ¬ V ∈ V | |
11 | 10 | pm2.21i 117 | . . . . . 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 322 | . . 3 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ (𝑆 × 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆) |
15 | 2, 14 | sylnbir 323 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆) |
16 | 1, 15 | pm2.61i 177 | 1 ⊢ ((𝐴𝐹𝐵)) ∈ 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 Vcvv 3410 〈cop 4442 × cxp 5402 dom cdm 5404 ((caov 42753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-int 4747 df-br 4927 df-opab 4989 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-res 5416 df-iota 6150 df-fun 6188 df-fv 6194 df-aiota 42721 df-dfat 42754 df-afv 42755 df-aov 42756 |
This theorem is referenced by: (None) |
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