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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 5104 | . . 3 ⊢ (𝐹(𝑉𝑂𝐸)𝑃 ↔ 〈𝐹, 𝑃〉 ∈ (𝑉𝑂𝐸)) | |
2 | ne0i 4292 | . . . 4 ⊢ (〈𝐹, 𝑃〉 ∈ (𝑉𝑂𝐸) → (𝑉𝑂𝐸) ≠ ∅) | |
3 | brovex.1 | . . . . . 6 ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) | |
4 | 3 | mpondm0 7586 | . . . . 5 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝑂𝐸) = ∅) |
5 | 4 | necon1ai 2969 | . . . 4 ⊢ ((𝑉𝑂𝐸) ≠ ∅ → (𝑉 ∈ V ∧ 𝐸 ∈ V)) |
6 | brovex.2 | . . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → Rel (𝑉𝑂𝐸)) | |
7 | brrelex12 5682 | . . . . . . 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 413 | . . . 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 216 | . 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 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 Vcvv 3443 ∅c0 4280 〈cop 4590 class class class wbr 5103 Rel wrel 5636 (class class class)co 7351 ∈ cmpo 7353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-xp 5637 df-rel 5638 df-dm 5641 df-iota 6445 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 |
This theorem is referenced by: brovmpoex 8146 |
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