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Mirrors > Home > MPE Home > Th. List > brovex | Structured version Visualization version GIF version |
Description: A binary relation of the value of an operation given by the maps-to notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.) |
Ref | Expression |
---|---|
brovex.1 | ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) |
brovex.2 | ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → Rel (𝑉𝑂𝐸)) |
Ref | Expression |
---|---|
brovex | ⊢ (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5149 | . . 3 ⊢ (𝐹(𝑉𝑂𝐸)𝑃 ↔ 〈𝐹, 𝑃〉 ∈ (𝑉𝑂𝐸)) | |
2 | ne0i 4347 | . . . 4 ⊢ (〈𝐹, 𝑃〉 ∈ (𝑉𝑂𝐸) → (𝑉𝑂𝐸) ≠ ∅) | |
3 | brovex.1 | . . . . . 6 ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) | |
4 | 3 | mpondm0 7673 | . . . . 5 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝑂𝐸) = ∅) |
5 | 4 | necon1ai 2966 | . . . 4 ⊢ ((𝑉𝑂𝐸) ≠ ∅ → (𝑉 ∈ V ∧ 𝐸 ∈ V)) |
6 | brovex.2 | . . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → Rel (𝑉𝑂𝐸)) | |
7 | brrelex12 5741 | . . . . . . 7 ⊢ ((Rel (𝑉𝑂𝐸) ∧ 𝐹(𝑉𝑂𝐸)𝑃) → (𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
8 | 6, 7 | sylan 580 | . . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝐹(𝑉𝑂𝐸)𝑃) → (𝐹 ∈ V ∧ 𝑃 ∈ V)) |
9 | id 22 | . . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) | |
10 | 8, 9 | syldan 591 | . . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝐹(𝑉𝑂𝐸)𝑃) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
11 | 10 | ex 412 | . . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))) |
12 | 2, 5, 11 | 3syl 18 | . . 3 ⊢ (〈𝐹, 𝑃〉 ∈ (𝑉𝑂𝐸) → (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))) |
13 | 1, 12 | sylbi 217 | . 2 ⊢ (𝐹(𝑉𝑂𝐸)𝑃 → (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))) |
14 | 13 | pm2.43i 52 | 1 ⊢ (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∅c0 4339 〈cop 4637 class class class wbr 5148 Rel wrel 5694 (class class class)co 7431 ∈ cmpo 7433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-dm 5699 df-iota 6516 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 |
This theorem is referenced by: brovmpoex 8247 |
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