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Theorem brovex 8228
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
StepHypRef Expression
1 df-br 5149 . . 3 (𝐹(𝑉𝑂𝐸)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (𝑉𝑂𝐸))
2 ne0i 4335 . . . 4 (⟨𝐹, 𝑃⟩ ∈ (𝑉𝑂𝐸) → (𝑉𝑂𝐸) ≠ ∅)
3 brovex.1 . . . . . 6 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶)
43mpondm0 7661 . . . . 5 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝑂𝐸) = ∅)
54necon1ai 2965 . . . 4 ((𝑉𝑂𝐸) ≠ ∅ → (𝑉 ∈ V ∧ 𝐸 ∈ V))
6 brovex.2 . . . . . . 7 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → Rel (𝑉𝑂𝐸))
7 brrelex12 5730 . . . . . . 7 ((Rel (𝑉𝑂𝐸) ∧ 𝐹(𝑉𝑂𝐸)𝑃) → (𝐹 ∈ V ∧ 𝑃 ∈ V))
86, 7sylan 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)))
108, 9syldan 590 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝐹(𝑉𝑂𝐸)𝑃) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
1110ex 412 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
122, 5, 113syl 18 . . 3 (⟨𝐹, 𝑃⟩ ∈ (𝑉𝑂𝐸) → (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
131, 12sylbi 216 . 2 (𝐹(𝑉𝑂𝐸)𝑃 → (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
1413pm2.43i 52 1 (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wne 2937  Vcvv 3471  c0 4323  cop 4635   class class class wbr 5148  Rel wrel 5683  (class class class)co 7420  cmpo 7422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-dm 5688  df-iota 6500  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425
This theorem is referenced by:  brovmpoex  8229
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