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| Mirrors > Home > MPE Home > Th. List > brovpreldm | Structured version Visualization version GIF version | ||
| Description: If a binary relation holds for the result of an operation, the operands are in the domain of the operation. (Contributed by AV, 31-Dec-2020.) |
| Ref | Expression |
|---|---|
| brovpreldm | ⊢ (𝐷(𝐵𝐴𝐶)𝐸 → 〈𝐵, 𝐶〉 ∈ dom 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-br 5144 | . 2 ⊢ (𝐷(𝐵𝐴𝐶)𝐸 ↔ 〈𝐷, 𝐸〉 ∈ (𝐵𝐴𝐶)) | |
| 2 | ne0i 4341 | . . 3 ⊢ (〈𝐷, 𝐸〉 ∈ (𝐵𝐴𝐶) → (𝐵𝐴𝐶) ≠ ∅) | |
| 3 | df-ov 7434 | . . . . 5 ⊢ (𝐵𝐴𝐶) = (𝐴‘〈𝐵, 𝐶〉) | |
| 4 | ndmfv 6941 | . . . . 5 ⊢ (¬ 〈𝐵, 𝐶〉 ∈ dom 𝐴 → (𝐴‘〈𝐵, 𝐶〉) = ∅) | |
| 5 | 3, 4 | eqtrid 2789 | . . . 4 ⊢ (¬ 〈𝐵, 𝐶〉 ∈ dom 𝐴 → (𝐵𝐴𝐶) = ∅) |
| 6 | 5 | necon1ai 2968 | . . 3 ⊢ ((𝐵𝐴𝐶) ≠ ∅ → 〈𝐵, 𝐶〉 ∈ dom 𝐴) |
| 7 | 2, 6 | syl 17 | . 2 ⊢ (〈𝐷, 𝐸〉 ∈ (𝐵𝐴𝐶) → 〈𝐵, 𝐶〉 ∈ dom 𝐴) |
| 8 | 1, 7 | sylbi 217 | 1 ⊢ (𝐷(𝐵𝐴𝐶)𝐸 → 〈𝐵, 𝐶〉 ∈ dom 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 〈cop 4632 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 |
| 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-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-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-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: bropopvvv 8115 bropfvvvv 8117 |
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