Theorem brovpreldm 8030

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 5087	. 2 ⊢ (𝐷(𝐵𝐴𝐶)𝐸 ↔ ⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶))
2		ne0i 4282	. . 3 ⊢ (⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶) → (𝐵𝐴𝐶) ≠ ∅)
3		df-ov 7361	. . . . 5 ⊢ (𝐵𝐴𝐶) = (𝐴‘⟨𝐵, 𝐶⟩)
4		ndmfv 6864	. . . . 5 ⊢ (¬ ⟨𝐵, 𝐶⟩ ∈ dom 𝐴 → (𝐴‘⟨𝐵, 𝐶⟩) = ∅)
5	3, 4	eqtrid 2784	. . . 4 ⊢ (¬ ⟨𝐵, 𝐶⟩ ∈ dom 𝐴 → (𝐵𝐴𝐶) = ∅)
6	5	necon1ai 2960	. . 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 2114   ≠ wne 2933  ∅c0 4274  ⟨cop 4574   class class class wbr 5086  dom cdm 5622  ‘cfv 6490  (class class class)co 7358

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5241  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-dm 5632  df-iota 6446  df-fv 6498  df-ov 7361

This theorem is referenced by:  bropopvvv  8031  bropfvvvv  8033