Theorem brovpreldm 7784

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 5067	. 2 ⊢ (𝐷(𝐵𝐴𝐶)𝐸 ↔ ⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶))
2		ne0i 4300	. . 3 ⊢ (⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶) → (𝐵𝐴𝐶) ≠ ∅)
3		df-ov 7159	. . . . 5 ⊢ (𝐵𝐴𝐶) = (𝐴‘⟨𝐵, 𝐶⟩)
4		ndmfv 6700	. . . . 5 ⊢ (¬ ⟨𝐵, 𝐶⟩ ∈ dom 𝐴 → (𝐴‘⟨𝐵, 𝐶⟩) = ∅)
5	3, 4	syl5eq 2868	. . . 4 ⊢ (¬ ⟨𝐵, 𝐶⟩ ∈ dom 𝐴 → (𝐵𝐴𝐶) = ∅)
6	5	necon1ai 3043	. . 3 ⊢ ((𝐵𝐴𝐶) ≠ ∅ → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)
7	2, 6	syl 17	. 2 ⊢ (⟨𝐷, 𝐸⟩ ∈ (𝐵𝐴𝐶) → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)
8	1, 7	sylbi 219	1 ⊢ (𝐷(𝐵𝐴𝐶)𝐸 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2114   ≠ wne 3016  ∅c0 4291  ⟨cop 4573   class class class wbr 5066  dom cdm 5555  ‘cfv 6355  (class class class)co 7156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-nul 5210  ax-pow 5266

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-dm 5565  df-iota 6314  df-fv 6363  df-ov 7159

This theorem is referenced by:  bropopvvv  7785  bropfvvvv  7787