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Theorem brovpreldm 7251
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.)
Assertion
Ref Expression
brovpreldm (𝐷(𝐵𝐴𝐶)𝐸 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)

Proof of Theorem brovpreldm
StepHypRef Expression
1 df-br 4652 . 2 (𝐷(𝐵𝐴𝐶)𝐸 ↔ ⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶))
2 ne0i 3919 . . 3 (⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶) → (𝐵𝐴𝐶) ≠ ∅)
3 df-ov 6650 . . . . 5 (𝐵𝐴𝐶) = (𝐴‘⟨𝐵, 𝐶⟩)
4 ndmfv 6216 . . . . 5 (¬ ⟨𝐵, 𝐶⟩ ∈ dom 𝐴 → (𝐴‘⟨𝐵, 𝐶⟩) = ∅)
53, 4syl5eq 2667 . . . 4 (¬ ⟨𝐵, 𝐶⟩ ∈ dom 𝐴 → (𝐵𝐴𝐶) = ∅)
65necon1ai 2820 . . 3 ((𝐵𝐴𝐶) ≠ ∅ → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)
72, 6syl 17 . 2 (⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶) → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)
81, 7sylbi 207 1 (𝐷(𝐵𝐴𝐶)𝐸 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 1989  wne 2793  c0 3913  cop 4181   class class class wbr 4651  dom cdm 5112  cfv 5886  (class class class)co 6647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-nul 4787  ax-pow 4841
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-dm 5122  df-iota 5849  df-fv 5894  df-ov 6650
This theorem is referenced by:  bropopvvv  7252  bropfvvvv  7254
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