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| Mirrors > Home > MPE Home > Th. List > Mathboxes > brovmptimex | Structured version Visualization version GIF version | ||
| Description: If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021.) |
| Ref | Expression |
|---|---|
| brovmptimex.mpt | ⊢ 𝐹 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻) |
| brovmptimex.br | ⊢ (𝜑 → 𝐴𝑅𝐵) |
| brovmptimex.ov | ⊢ (𝜑 → 𝑅 = (𝐶𝐹𝐷)) |
| Ref | Expression |
|---|---|
| brovmptimex | ⊢ (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | brovmptimex.ov | . . 3 ⊢ (𝜑 → 𝑅 = (𝐶𝐹𝐷)) | |
| 2 | brovmptimex.br | . . 3 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
| 3 | 1, 2 | breqdi 5158 | . 2 ⊢ (𝜑 → 𝐴(𝐶𝐹𝐷)𝐵) |
| 4 | brne0 5193 | . 2 ⊢ (𝐴(𝐶𝐹𝐷)𝐵 → (𝐶𝐹𝐷) ≠ ∅) | |
| 5 | brovmptimex.mpt | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻) | |
| 6 | 5 | reldmmpo 7567 | . . . 4 ⊢ Rel dom 𝐹 |
| 7 | 6 | ovprc 7469 | . . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶𝐹𝐷) = ∅) |
| 8 | 7 | necon1ai 2968 | . 2 ⊢ ((𝐶𝐹𝐷) ≠ ∅ → (𝐶 ∈ V ∧ 𝐷 ∈ V)) |
| 9 | 3, 4, 8 | 3syl 18 | 1 ⊢ (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 class class class wbr 5143 (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 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 |
| This theorem is referenced by: brovmptimex1 44041 brovmptimex2 44042 |
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