Theorem brovex 8206

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 5149	. . 3 ⊢ (𝐹(𝑉𝑂𝐸)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (𝑉𝑂𝐸))
2		ne0i 4334	. . . 4 ⊢ (⟨𝐹, 𝑃⟩ ∈ (𝑉𝑂𝐸) → (𝑉𝑂𝐸) ≠ ∅)
3		brovex.1	. . . . . 6 ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶)
4	3	mpondm0 7646	. . . . 5 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝑂𝐸) = ∅)
5	4	necon1ai 2968	. . . 4 ⊢ ((𝑉𝑂𝐸) ≠ ∅ → (𝑉 ∈ V ∧ 𝐸 ∈ V))
6		brovex.2	. . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → Rel (𝑉𝑂𝐸))
7		brrelex12 5728	. . . . . . 7 ⊢ ((Rel (𝑉𝑂𝐸) ∧ 𝐹(𝑉𝑂𝐸)𝑃) → (𝐹 ∈ V ∧ 𝑃 ∈ V))
8	6, 7	sylan 580	. . . . . 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 591	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝐹(𝑉𝑂𝐸)𝑃) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
11	10	ex 413	. . . 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 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  Vcvv 3474  ∅c0 4322  ⟨cop 4634   class class class wbr 5148  Rel wrel 5681  (class class class)co 7408   ∈ cmpo 7410

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-dm 5686  df-iota 6495  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413

This theorem is referenced by:  brovmpoex  8207