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Mirrors > Home > MPE Home > Th. List > 2mpo0 | Structured version Visualization version GIF version |
Description: If the operation value of the operation value of two nested maps-to notation is not empty, all involved arguments belong to the corresponding base classes of the maps-to notations. (Contributed by AV, 21-May-2021.) |
Ref | Expression |
---|---|
2mpo0.o | ⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) |
2mpo0.u | ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) = (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)) |
Ref | Expression |
---|---|
2mpo0 | ⊢ (¬ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ianor 979 | . 2 ⊢ (¬ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) ↔ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∨ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷))) | |
2 | 2mpo0.o | . . . . . 6 ⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) | |
3 | 2 | mpondm0 7510 | . . . . 5 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) = ∅) |
4 | 3 | oveqd 7292 | . . . 4 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆∅𝑇)) |
5 | 0ov 7312 | . . . 4 ⊢ (𝑆∅𝑇) = ∅ | |
6 | 4, 5 | eqtrdi 2794 | . . 3 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅) |
7 | notnotb 315 | . . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ↔ ¬ ¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) | |
8 | 2mpo0.u | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) = (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)) | |
9 | 8 | adantr 481 | . . . . . 6 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑋𝑂𝑌) = (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)) |
10 | 9 | oveqd 7292 | . . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆(𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)𝑇)) |
11 | eqid 2738 | . . . . . . 7 ⊢ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹) = (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹) | |
12 | 11 | mpondm0 7510 | . . . . . 6 ⊢ (¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷) → (𝑆(𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)𝑇) = ∅) |
13 | 12 | adantl 482 | . . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)𝑇) = ∅) |
14 | 10, 13 | eqtrd 2778 | . . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅) |
15 | 7, 14 | sylanbr 582 | . . 3 ⊢ ((¬ ¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅) |
16 | 6, 15 | jaoi3 1058 | . 2 ⊢ ((¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∨ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅) |
17 | 1, 16 | sylbi 216 | 1 ⊢ (¬ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 ∅c0 4256 (class class class)co 7275 ∈ cmpo 7277 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-xp 5595 df-dm 5599 df-iota 6391 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 |
This theorem is referenced by: wwlksnon0 28219 |
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