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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 4654 | . 2 ⊢ (𝐷(𝐵𝐴𝐶)𝐸 ↔ 〈𝐷, 𝐸〉 ∈ (𝐵𝐴𝐶)) | |
2 | ne0i 3921 | . . 3 ⊢ (〈𝐷, 𝐸〉 ∈ (𝐵𝐴𝐶) → (𝐵𝐴𝐶) ≠ ∅) | |
3 | df-ov 6653 | . . . . 5 ⊢ (𝐵𝐴𝐶) = (𝐴‘〈𝐵, 𝐶〉) | |
4 | ndmfv 6218 | . . . . 5 ⊢ (¬ 〈𝐵, 𝐶〉 ∈ dom 𝐴 → (𝐴‘〈𝐵, 𝐶〉) = ∅) | |
5 | 3, 4 | syl5eq 2668 | . . . 4 ⊢ (¬ 〈𝐵, 𝐶〉 ∈ dom 𝐴 → (𝐵𝐴𝐶) = ∅) |
6 | 5 | necon1ai 2821 | . . 3 ⊢ ((𝐵𝐴𝐶) ≠ ∅ → 〈𝐵, 𝐶〉 ∈ dom 𝐴) |
7 | 2, 6 | syl 17 | . 2 ⊢ (〈𝐷, 𝐸〉 ∈ (𝐵𝐴𝐶) → 〈𝐵, 𝐶〉 ∈ dom 𝐴) |
8 | 1, 7 | sylbi 207 | 1 ⊢ (𝐷(𝐵𝐴𝐶)𝐸 → 〈𝐵, 𝐶〉 ∈ dom 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 〈cop 4183 class class class wbr 4653 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 ax-pow 4843 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-dm 5124 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: bropopvvv 7255 bropfvvvv 7257 |
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