Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  brovpreldm Structured version   Visualization version   GIF version

Theorem brovpreldm 7777
 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 5034 . 2 (𝐷(𝐵𝐴𝐶)𝐸 ↔ ⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶))
2 ne0i 4252 . . 3 (⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶) → (𝐵𝐴𝐶) ≠ ∅)
3 df-ov 7145 . . . . 5 (𝐵𝐴𝐶) = (𝐴‘⟨𝐵, 𝐶⟩)
4 ndmfv 6682 . . . . 5 (¬ ⟨𝐵, 𝐶⟩ ∈ dom 𝐴 → (𝐴‘⟨𝐵, 𝐶⟩) = ∅)
53, 4syl5eq 2845 . . . 4 (¬ ⟨𝐵, 𝐶⟩ ∈ dom 𝐴 → (𝐵𝐴𝐶) = ∅)
65necon1ai 3014 . . 3 ((𝐵𝐴𝐶) ≠ ∅ → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)
72, 6syl 17 . 2 (⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶) → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)
81, 7sylbi 220 1 (𝐷(𝐵𝐴𝐶)𝐸 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2111   ≠ wne 2987  ∅c0 4245  ⟨cop 4533   class class class wbr 5033  dom cdm 5522  ‘cfv 6329  (class class class)co 7142 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-br 5034  df-dm 5532  df-iota 6288  df-fv 6337  df-ov 7145 This theorem is referenced by:  bropopvvv  7778  bropfvvvv  7780
 Copyright terms: Public domain W3C validator