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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 981 | . 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 7647 | . . . . 5 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) = ∅) |
4 | 3 | oveqd 7426 | . . . 4 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆∅𝑇)) |
5 | 0ov 7446 | . . . 4 ⊢ (𝑆∅𝑇) = ∅ | |
6 | 4, 5 | eqtrdi 2789 | . . 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 482 | . . . . . 6 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑋𝑂𝑌) = (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)) |
10 | 9 | oveqd 7426 | . . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆(𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)𝑇)) |
11 | eqid 2733 | . . . . . . 7 ⊢ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹) = (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹) | |
12 | 11 | mpondm0 7647 | . . . . . 6 ⊢ (¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷) → (𝑆(𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)𝑇) = ∅) |
13 | 12 | adantl 483 | . . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)𝑇) = ∅) |
14 | 10, 13 | eqtrd 2773 | . . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅) |
15 | 7, 14 | sylanbr 583 | . . 3 ⊢ ((¬ ¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅) |
16 | 6, 15 | jaoi3 1060 | . 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 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∅c0 4323 (class class class)co 7409 ∈ cmpo 7411 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-dm 5687 df-iota 6496 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 |
This theorem is referenced by: wwlksnon0 29108 |
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