Theorem brovpreldm 7423

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 4805	. 2 ⊢ (𝐷(𝐵𝐴𝐶)𝐸 ↔ ⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶))
2		ne0i 4064	. . 3 ⊢ (⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶) → (𝐵𝐴𝐶) ≠ ∅)
3		df-ov 6817	. . . . 5 ⊢ (𝐵𝐴𝐶) = (𝐴‘⟨𝐵, 𝐶⟩)
4		ndmfv 6380	. . . . 5 ⊢ (¬ ⟨𝐵, 𝐶⟩ ∈ dom 𝐴 → (𝐴‘⟨𝐵, 𝐶⟩) = ∅)
5	3, 4	syl5eq 2806	. . . 4 ⊢ (¬ ⟨𝐵, 𝐶⟩ ∈ dom 𝐴 → (𝐵𝐴𝐶) = ∅)
6	5	necon1ai 2959	. . 3 ⊢ ((𝐵𝐴𝐶) ≠ ∅ → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)
7	2, 6	syl 17	. 2 ⊢ (⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶) → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)
8	1, 7	sylbi 207	1 ⊢ (𝐷(𝐵𝐴𝐶)𝐸 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2139   ≠ wne 2932  ∅c0 4058  ⟨cop 4327   class class class wbr 4804  dom cdm 5266  ‘cfv 6049  (class class class)co 6814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941  ax-pow 4992

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-dm 5276  df-iota 6012  df-fv 6057  df-ov 6817

This theorem is referenced by:  bropopvvv  7424  bropfvvvv  7426