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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 7551 | . . . 4 ⊢ Rel dom 𝐹 |
7 | 6 | ovprc 7453 | . . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶𝐹𝐷) = ∅) |
8 | 7 | necon1ai 2958 | . 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 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 Vcvv 3463 ∅c0 4318 class class class wbr 5143 (class class class)co 7415 ∈ cmpo 7417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3943 df-un 3945 df-ss 3957 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-xp 5678 df-rel 5679 df-dm 5682 df-iota 6494 df-fv 6550 df-ov 7418 df-oprab 7419 df-mpo 7420 |
This theorem is referenced by: brovmptimex1 43522 brovmptimex2 43523 |
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