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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 5094 | . 2 ⊢ (𝜑 → 𝐴(𝐶𝐹𝐷)𝐵) |
4 | brne0 5129 | . 2 ⊢ (𝐴(𝐶𝐹𝐷)𝐵 → (𝐶𝐹𝐷) ≠ ∅) | |
5 | brovmptimex.mpt | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻) | |
6 | 5 | reldmmpo 7402 | . . . 4 ⊢ Rel dom 𝐹 |
7 | 6 | ovprc 7309 | . . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶𝐹𝐷) = ∅) |
8 | 7 | necon1ai 2973 | . 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 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 Vcvv 3431 ∅c0 4262 class class class wbr 5079 (class class class)co 7271 ∈ cmpo 7273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-xp 5596 df-rel 5597 df-dm 5600 df-iota 6390 df-fv 6440 df-ov 7274 df-oprab 7275 df-mpo 7276 |
This theorem is referenced by: brovmptimex1 41608 brovmptimex2 41609 |
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