Theorem brovex 8203

Description: A binary relation of the value of an operation given by the maps-to notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)

Hypotheses

Ref	Expression
brovex.1	⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶)
brovex.2	⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → Rel (𝑉𝑂𝐸))

Assertion

Ref	Expression
brovex	⊢ (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑃(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑂(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem brovex

Step	Hyp	Ref	Expression
1		df-br 5140	. . 3 ⊢ (𝐹(𝑉𝑂𝐸)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (𝑉𝑂𝐸))
2		ne0i 4327	. . . 4 ⊢ (⟨𝐹, 𝑃⟩ ∈ (𝑉𝑂𝐸) → (𝑉𝑂𝐸) ≠ ∅)
3		brovex.1	. . . . . 6 ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶)
4	3	mpondm0 7641	. . . . 5 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝑂𝐸) = ∅)
5	4	necon1ai 2960	. . . 4 ⊢ ((𝑉𝑂𝐸) ≠ ∅ → (𝑉 ∈ V ∧ 𝐸 ∈ V))
6		brovex.2	. . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → Rel (𝑉𝑂𝐸))
7		brrelex12 5719	. . . . . . 7 ⊢ ((Rel (𝑉𝑂𝐸) ∧ 𝐹(𝑉𝑂𝐸)𝑃) → (𝐹 ∈ V ∧ 𝑃 ∈ V))
8	6, 7	sylan 579	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝐹(𝑉𝑂𝐸)𝑃) → (𝐹 ∈ V ∧ 𝑃 ∈ V))
9		id 22	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
10	8, 9	syldan 590	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝐹(𝑉𝑂𝐸)𝑃) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
11	10	ex 412	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
12	2, 5, 11	3syl 18	. . 3 ⊢ (⟨𝐹, 𝑃⟩ ∈ (𝑉𝑂𝐸) → (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
13	1, 12	sylbi 216	. 2 ⊢ (𝐹(𝑉𝑂𝐸)𝑃 → (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
14	13	pm2.43i 52	1 ⊢ (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ≠ wne 2932  Vcvv 3466  ∅c0 4315  ⟨cop 4627   class class class wbr 5139  Rel wrel 5672  (class class class)co 7402   ∈ cmpo 7404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-dm 5677  df-iota 6486  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407

This theorem is referenced by:  brovmpoex  8204