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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 7335 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 5593 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) | |
3 | ndmaov.1 | . . . . . 6 ⊢ dom 𝐹 = (𝑆 × 𝑆) | |
4 | 3 | eqcomi 2832 | . . . . 5 ⊢ (𝑆 × 𝑆) = dom 𝐹 |
5 | 4 | eleq2i 2906 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑆 × 𝑆) ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
6 | ndmaovcl.3 | . . . . 5 ⊢ ((𝐴𝐹𝐵)) ∈ V | |
7 | ndmaov 43389 | . . . . 5 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V) | |
8 | eleq1 2902 | . . . . . . 7 ⊢ ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V ↔ V ∈ V)) | |
9 | 8 | biimpd 231 | . . . . . 6 ⊢ ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V → V ∈ V)) |
10 | vprc 5221 | . . . . . . 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 332 | . . 3 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ (𝑆 × 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆) |
15 | 2, 14 | sylnbir 333 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆) |
16 | 1, 15 | pm2.61i 184 | 1 ⊢ ((𝐴𝐹𝐵)) ∈ 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 〈cop 4575 × cxp 5555 dom cdm 5557 ((caov 43324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-res 5569 df-iota 6316 df-fun 6359 df-fv 6365 df-aiota 43292 df-dfat 43325 df-afv 43326 df-aov 43327 |
This theorem is referenced by: (None) |
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