Theorem brovpreldm 8068

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

Step	Hyp	Ref	Expression
1		df-br 5108	. 2 ⊢ (𝐷(𝐵𝐴𝐶)𝐸 ↔ ⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶))
2		ne0i 4304	. . 3 ⊢ (⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶) → (𝐵𝐴𝐶) ≠ ∅)
3		df-ov 7390	. . . . 5 ⊢ (𝐵𝐴𝐶) = (𝐴‘⟨𝐵, 𝐶⟩)
4		ndmfv 6893	. . . . 5 ⊢ (¬ ⟨𝐵, 𝐶⟩ ∈ dom 𝐴 → (𝐴‘⟨𝐵, 𝐶⟩) = ∅)
5	3, 4	eqtrid 2776	. . . 4 ⊢ (¬ ⟨𝐵, 𝐶⟩ ∈ dom 𝐴 → (𝐵𝐴𝐶) = ∅)
6	5	necon1ai 2952	. . 3 ⊢ ((𝐵𝐴𝐶) ≠ ∅ → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)
7	2, 6	syl 17	. 2 ⊢ (⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶) → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)
8	1, 7	sylbi 217	1 ⊢ (𝐷(𝐵𝐴𝐶)𝐸 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2109   ≠ wne 2925  ∅c0 4296  ⟨cop 4595   class class class wbr 5107  dom cdm 5638  ‘cfv 6511  (class class class)co 7387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-dm 5648  df-iota 6464  df-fv 6519  df-ov 7390

This theorem is referenced by:  bropopvvv  8069  bropfvvvv  8071