Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5058 | . 2 ⊢ (𝐷(𝐵𝐴𝐶)𝐸 ↔ 〈𝐷, 𝐸〉 ∈ (𝐵𝐴𝐶)) | |
2 | ne0i 4298 | . . 3 ⊢ (〈𝐷, 𝐸〉 ∈ (𝐵𝐴𝐶) → (𝐵𝐴𝐶) ≠ ∅) | |
3 | df-ov 7151 | . . . . 5 ⊢ (𝐵𝐴𝐶) = (𝐴‘〈𝐵, 𝐶〉) | |
4 | ndmfv 6693 | . . . . 5 ⊢ (¬ 〈𝐵, 𝐶〉 ∈ dom 𝐴 → (𝐴‘〈𝐵, 𝐶〉) = ∅) | |
5 | 3, 4 | syl5eq 2866 | . . . 4 ⊢ (¬ 〈𝐵, 𝐶〉 ∈ dom 𝐴 → (𝐵𝐴𝐶) = ∅) |
6 | 5 | necon1ai 3041 | . . 3 ⊢ ((𝐵𝐴𝐶) ≠ ∅ → 〈𝐵, 𝐶〉 ∈ dom 𝐴) |
7 | 2, 6 | syl 17 | . 2 ⊢ (〈𝐷, 𝐸〉 ∈ (𝐵𝐴𝐶) → 〈𝐵, 𝐶〉 ∈ dom 𝐴) |
8 | 1, 7 | sylbi 219 | 1 ⊢ (𝐷(𝐵𝐴𝐶)𝐸 → 〈𝐵, 𝐶〉 ∈ dom 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2108 ≠ wne 3014 ∅c0 4289 〈cop 4565 class class class wbr 5057 dom cdm 5548 ‘cfv 6348 (class class class)co 7148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-nul 5201 ax-pow 5257 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-dm 5558 df-iota 6307 df-fv 6356 df-ov 7151 |
This theorem is referenced by: bropopvvv 7777 bropfvvvv 7779 |
Copyright terms: Public domain | W3C validator |