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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 4584 | . 2 ⊢ (𝐷(𝐵𝐴𝐶)𝐸 ↔ 〈𝐷, 𝐸〉 ∈ (𝐵𝐴𝐶)) | |
2 | ne0i 3880 | . . 3 ⊢ (〈𝐷, 𝐸〉 ∈ (𝐵𝐴𝐶) → (𝐵𝐴𝐶) ≠ ∅) | |
3 | df-ov 6552 | . . . . 5 ⊢ (𝐵𝐴𝐶) = (𝐴‘〈𝐵, 𝐶〉) | |
4 | ndmfv 6128 | . . . . 5 ⊢ (¬ 〈𝐵, 𝐶〉 ∈ dom 𝐴 → (𝐴‘〈𝐵, 𝐶〉) = ∅) | |
5 | 3, 4 | syl5eq 2656 | . . . 4 ⊢ (¬ 〈𝐵, 𝐶〉 ∈ dom 𝐴 → (𝐵𝐴𝐶) = ∅) |
6 | 5 | necon1ai 2809 | . . 3 ⊢ ((𝐵𝐴𝐶) ≠ ∅ → 〈𝐵, 𝐶〉 ∈ dom 𝐴) |
7 | 2, 6 | syl 17 | . 2 ⊢ (〈𝐷, 𝐸〉 ∈ (𝐵𝐴𝐶) → 〈𝐵, 𝐶〉 ∈ dom 𝐴) |
8 | 1, 7 | sylbi 206 | 1 ⊢ (𝐷(𝐵𝐴𝐶)𝐸 → 〈𝐵, 𝐶〉 ∈ dom 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 〈cop 4131 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 ax-pow 4769 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-dm 5048 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: bropopvvv 7142 bropfvvvv 7144 |
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