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Theorem bropopvvv 7498
Description: If a binary relation holds for the result of an operation which is a result of an operation, the involved classes are sets. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Proof shortened by AV, 3-Jan-2021.)
Hypotheses
Ref Expression
bropopvvv.o 𝑂 = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜑}))
bropopvvv.p ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜑𝜓))
bropopvvv.oo (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉𝑂𝐸)𝐵) = {⟨𝑓, 𝑝⟩ ∣ 𝜃})
Assertion
Ref Expression
bropopvvv (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))
Distinct variable groups:   𝐸,𝑎,𝑏,𝑒,𝑓,𝑝,𝑣   𝑉,𝑎,𝑏,𝑒,𝑓,𝑝,𝑣   𝜓,𝑒,𝑣
Allowed substitution hints:   𝜑(𝑣,𝑒,𝑓,𝑝,𝑎,𝑏)   𝜓(𝑓,𝑝,𝑎,𝑏)   𝜃(𝑣,𝑒,𝑓,𝑝,𝑎,𝑏)   𝐴(𝑣,𝑒,𝑓,𝑝,𝑎,𝑏)   𝐵(𝑣,𝑒,𝑓,𝑝,𝑎,𝑏)   𝑃(𝑣,𝑒,𝑓,𝑝,𝑎,𝑏)   𝐹(𝑣,𝑒,𝑓,𝑝,𝑎,𝑏)   𝑂(𝑣,𝑒,𝑓,𝑝,𝑎,𝑏)

Proof of Theorem bropopvvv
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 brovpreldm 7497 . . 3 (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸))
2 simpl 470 . . . . . . . . 9 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑣 = 𝑉)
3 bropopvvv.p . . . . . . . . . 10 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜑𝜓))
43opabbidv 4921 . . . . . . . . 9 ((𝑣 = 𝑉𝑒 = 𝐸) → {⟨𝑓, 𝑝⟩ ∣ 𝜑} = {⟨𝑓, 𝑝⟩ ∣ 𝜓})
52, 2, 4mpt2eq123dv 6956 . . . . . . . 8 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜑}) = (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}))
6 bropopvvv.o . . . . . . . 8 𝑂 = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜑}))
75, 6ovmpt2ga 7029 . . . . . . 7 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) → (𝑉𝑂𝐸) = (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}))
87dmeqd 5540 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) → dom (𝑉𝑂𝐸) = dom (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}))
98eleq2d 2882 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) ↔ ⟨𝐴, 𝐵⟩ ∈ dom (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓})))
10 dmoprabss 6981 . . . . . . . 8 dom {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ((𝑎𝑉𝑏𝑉) ∧ 𝑐 = {⟨𝑓, 𝑝⟩ ∣ 𝜓})} ⊆ (𝑉 × 𝑉)
1110sseli 3805 . . . . . . 7 (⟨𝐴, 𝐵⟩ ∈ dom {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ((𝑎𝑉𝑏𝑉) ∧ 𝑐 = {⟨𝑓, 𝑝⟩ ∣ 𝜓})} → ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉))
12 opelxp 5359 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉) ↔ (𝐴𝑉𝐵𝑉))
13 df-br 4856 . . . . . . . . . . . . 13 (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (𝐴(𝑉𝑂𝐸)𝐵))
14 ne0i 4133 . . . . . . . . . . . . . 14 (⟨𝐹, 𝑃⟩ ∈ (𝐴(𝑉𝑂𝐸)𝐵) → (𝐴(𝑉𝑂𝐸)𝐵) ≠ ∅)
15 bropopvvv.oo . . . . . . . . . . . . . . . . . . . 20 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉𝑂𝐸)𝐵) = {⟨𝑓, 𝑝⟩ ∣ 𝜃})
1615breqd 4866 . . . . . . . . . . . . . . . . . . 19 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃𝐹{⟨𝑓, 𝑝⟩ ∣ 𝜃}𝑃))
17 brabv 6938 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹{⟨𝑓, 𝑝⟩ ∣ 𝜃}𝑃 → (𝐹 ∈ V ∧ 𝑃 ∈ V))
1817anim2i 605 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝐹{⟨𝑓, 𝑝⟩ ∣ 𝜃}𝑃) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
1918ex 399 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝐹{⟨𝑓, 𝑝⟩ ∣ 𝜃}𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
2019adantr 468 . . . . . . . . . . . . . . . . . . 19 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹{⟨𝑓, 𝑝⟩ ∣ 𝜃}𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
2116, 20sylbid 231 . . . . . . . . . . . . . . . . . 18 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
2221ex 399 . . . . . . . . . . . . . . . . 17 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝐴𝑉𝐵𝑉) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))))
2322com23 86 . . . . . . . . . . . . . . . 16 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝐴𝑉𝐵𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))))
2423a1d 25 . . . . . . . . . . . . . . 15 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝐴(𝑉𝑂𝐸)𝐵) ≠ ∅ → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝐴𝑉𝐵𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))))
256mpt2ndm0 7114 . . . . . . . . . . . . . . . 16 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝑂𝐸) = ∅)
26 df-ov 6886 . . . . . . . . . . . . . . . . . 18 (𝐴(𝑉𝑂𝐸)𝐵) = ((𝑉𝑂𝐸)‘⟨𝐴, 𝐵⟩)
27 fveq1 6416 . . . . . . . . . . . . . . . . . 18 ((𝑉𝑂𝐸) = ∅ → ((𝑉𝑂𝐸)‘⟨𝐴, 𝐵⟩) = (∅‘⟨𝐴, 𝐵⟩))
2826, 27syl5eq 2863 . . . . . . . . . . . . . . . . 17 ((𝑉𝑂𝐸) = ∅ → (𝐴(𝑉𝑂𝐸)𝐵) = (∅‘⟨𝐴, 𝐵⟩))
29 0fv 6456 . . . . . . . . . . . . . . . . 17 (∅‘⟨𝐴, 𝐵⟩) = ∅
3028, 29syl6eq 2867 . . . . . . . . . . . . . . . 16 ((𝑉𝑂𝐸) = ∅ → (𝐴(𝑉𝑂𝐸)𝐵) = ∅)
31 eqneqall 3000 . . . . . . . . . . . . . . . 16 ((𝐴(𝑉𝑂𝐸)𝐵) = ∅ → ((𝐴(𝑉𝑂𝐸)𝐵) ≠ ∅ → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝐴𝑉𝐵𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))))
3225, 30, 313syl 18 . . . . . . . . . . . . . . 15 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝐴(𝑉𝑂𝐸)𝐵) ≠ ∅ → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝐴𝑉𝐵𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))))
3324, 32pm2.61i 176 . . . . . . . . . . . . . 14 ((𝐴(𝑉𝑂𝐸)𝐵) ≠ ∅ → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝐴𝑉𝐵𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))))
3414, 33syl 17 . . . . . . . . . . . . 13 (⟨𝐹, 𝑃⟩ ∈ (𝐴(𝑉𝑂𝐸)𝐵) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝐴𝑉𝐵𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))))
3513, 34sylbi 208 . . . . . . . . . . . 12 (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝐴𝑉𝐵𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))))
3635pm2.43i 52 . . . . . . . . . . 11 (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝐴𝑉𝐵𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
3736com12 32 . . . . . . . . . 10 ((𝐴𝑉𝐵𝑉) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
3837anc2ri 548 . . . . . . . . 9 ((𝐴𝑉𝐵𝑉) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐴𝑉𝐵𝑉))))
39 df-3an 1102 . . . . . . . . 9 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ↔ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐴𝑉𝐵𝑉)))
4038, 39syl6ibr 243 . . . . . . . 8 ((𝐴𝑉𝐵𝑉) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉))))
4112, 40sylbi 208 . . . . . . 7 (⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉))))
4211, 41syl 17 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ dom {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ((𝑎𝑉𝑏𝑉) ∧ 𝑐 = {⟨𝑓, 𝑝⟩ ∣ 𝜓})} → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉))))
43 df-mpt2 6888 . . . . . . 7 (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) = {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ((𝑎𝑉𝑏𝑉) ∧ 𝑐 = {⟨𝑓, 𝑝⟩ ∣ 𝜓})}
4443dmeqi 5539 . . . . . 6 dom (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) = dom {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ((𝑎𝑉𝑏𝑉) ∧ 𝑐 = {⟨𝑓, 𝑝⟩ ∣ 𝜓})}
4542, 44eleq2s 2914 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ dom (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉))))
469, 45syl6bi 244 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))))
47 3ianor 1125 . . . . 5 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) ↔ (¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V ∨ ¬ (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V))
48 df-3or 1101 . . . . . 6 ((¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V ∨ ¬ (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) ↔ ((¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V) ∨ ¬ (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V))
49 ianor 995 . . . . . . . 8 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) ↔ (¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V))
5025dmeqd 5540 . . . . . . . . . . 11 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → dom (𝑉𝑂𝐸) = dom ∅)
5150eleq2d 2882 . . . . . . . . . 10 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) ↔ ⟨𝐴, 𝐵⟩ ∈ dom ∅))
52 dm0 5553 . . . . . . . . . . 11 dom ∅ = ∅
5352eleq2i 2888 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩ ∈ dom ∅ ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
5451, 53syl6bb 278 . . . . . . . . 9 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) ↔ ⟨𝐴, 𝐵⟩ ∈ ∅))
55 noel 4131 . . . . . . . . . 10 ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
5655pm2.21i 117 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ ∅ → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉))))
5754, 56syl6bi 244 . . . . . . . 8 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))))
5849, 57sylbir 226 . . . . . . 7 ((¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))))
59 anor 996 . . . . . . . . 9 ((𝑉 ∈ V ∧ 𝐸 ∈ V) ↔ ¬ (¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V))
60 id 22 . . . . . . . . . . . . 13 (𝑉 ∈ V → 𝑉 ∈ V)
6160ancri 541 . . . . . . . . . . . 12 (𝑉 ∈ V → (𝑉 ∈ V ∧ 𝑉 ∈ V))
6261adantr 468 . . . . . . . . . . 11 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ∈ V ∧ 𝑉 ∈ V))
63 mpt2exga 7488 . . . . . . . . . . 11 ((𝑉 ∈ V ∧ 𝑉 ∈ V) → (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V)
6462, 63syl 17 . . . . . . . . . 10 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V)
6564pm2.24d 148 . . . . . . . . 9 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (¬ (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉))))))
6659, 65sylbir 226 . . . . . . . 8 (¬ (¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V) → (¬ (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉))))))
6766imp 395 . . . . . . 7 ((¬ (¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V) ∧ ¬ (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))))
6858, 67jaoi3 1076 . . . . . 6 (((¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V) ∨ ¬ (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))))
6948, 68sylbi 208 . . . . 5 ((¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V ∨ ¬ (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))))
7047, 69sylbi 208 . . . 4 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))))
7146, 70pm2.61i 176 . . 3 (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉))))
721, 71syl 17 . 2 (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉))))
7372pm2.43i 52 1 (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865  w3o 1099  w3a 1100   = wceq 1637  wcel 2157  wne 2989  Vcvv 3402  c0 4127  cop 4387   class class class wbr 4855  {copab 4917   × cxp 5322  dom cdm 5324  cfv 6110  (class class class)co 6883  {coprab 6884  cmpt2 6885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7188
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6073  df-fun 6112  df-fn 6113  df-f 6114  df-f1 6115  df-fo 6116  df-f1o 6117  df-fv 6118  df-ov 6886  df-oprab 6887  df-mpt2 6888  df-1st 7407  df-2nd 7408
This theorem is referenced by: (None)
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