Theorem bropopvvv 8091

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 ↦ (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜑}))
bropopvvv.p	⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜑 ↔ 𝜓))
bropopvvv.oo	⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑉𝑂𝐸)𝐵) = {⟨𝑓, 𝑝⟩ ∣ 𝜃})

Assertion

Ref	Expression
bropopvvv	⊢ (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))

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.

Step	Hyp	Ref	Expression
1		brovpreldm 8090	. . 3 ⊢ (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸))
2		simpl 488	. . . . . . . . 9 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑣 = 𝑉)
3		bropopvvv.p	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜑 ↔ 𝜓))
4	3	opabbidv 5179	. . . . . . . . 9 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → {⟨𝑓, 𝑝⟩ ∣ 𝜑} = {⟨𝑓, 𝑝⟩ ∣ 𝜓})
5	2, 2, 4	mpoeq123dv 7494	. . . . . . . 8 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜑}) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}))
6		bropopvvv.o	. . . . . . . 8 ⊢ 𝑂 = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜑}))
7	5, 6	ovmpoga 7573	. . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) → (𝑉𝑂𝐸) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}))
8	7	dmeqd 5897	. . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) → dom (𝑉𝑂𝐸) = dom (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}))
9	8	eleq2d 2851	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) ↔ ⟨𝐴, 𝐵⟩ ∈ dom (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓})))
10		dmoprabss 7523	. . . . . . . 8 ⊢ dom {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑐 = {⟨𝑓, 𝑝⟩ ∣ 𝜓})} ⊆ (𝑉 × 𝑉)
11	10	sseli 3934	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑐 = {⟨𝑓, 𝑝⟩ ∣ 𝜓})} → ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉))
12		opelxp 5699	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉) ↔ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
13		df-br 5112	. . . . . . . . . . . . 13 ⊢ (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (𝐴(𝑉𝑂𝐸)𝐵))
14		ne0i 4294	. . . . . . . . . . . . . 14 ⊢ (⟨𝐹, 𝑃⟩ ∈ (𝐴(𝑉𝑂𝐸)𝐵) → (𝐴(𝑉𝑂𝐸)𝐵) ≠ ∅)
15		bropopvvv.oo	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑉𝑂𝐸)𝐵) = {⟨𝑓, 𝑝⟩ ∣ 𝜃})
16	15	breqd 5122	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 ↔ 𝐹{⟨𝑓, 𝑝⟩ ∣ 𝜃}𝑃))
17		brabv 5553	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹{⟨𝑓, 𝑝⟩ ∣ 𝜃}𝑃 → (𝐹 ∈ V ∧ 𝑃 ∈ V))
18	17	anim2i 629	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝐹{⟨𝑓, 𝑝⟩ ∣ 𝜃}𝑃) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
19	18	ex 418	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝐹{⟨𝑓, 𝑝⟩ ∣ 𝜃}𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
20	19	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐹{⟨𝑓, 𝑝⟩ ∣ 𝜃}𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
21	16, 20	sylbid 243	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
22	21	ex 418	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))))
23	22	com23 87	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))))
24	23	a1d 26	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝐴(𝑉𝑂𝐸)𝐵) ≠ ∅ → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))))
25	6	mpondm0 7660	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝑂𝐸) = ∅)
26		df-ov 7422	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴(𝑉𝑂𝐸)𝐵) = ((𝑉𝑂𝐸)‘⟨𝐴, 𝐵⟩)
27		fveq1 6884	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑉𝑂𝐸) = ∅ → ((𝑉𝑂𝐸)‘⟨𝐴, 𝐵⟩) = (∅‘⟨𝐴, 𝐵⟩))
28	26, 27	eqtrid 2812	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑉𝑂𝐸) = ∅ → (𝐴(𝑉𝑂𝐸)𝐵) = (∅‘⟨𝐴, 𝐵⟩))
29		0fv 6926	. . . . . . . . . . . . . . . . 17 ⊢ (∅‘⟨𝐴, 𝐵⟩) = ∅
30	28, 29	eqtrdi 2816	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉𝑂𝐸) = ∅ → (𝐴(𝑉𝑂𝐸)𝐵) = ∅)
31		eqneqall 2971	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴(𝑉𝑂𝐸)𝐵) = ∅ → ((𝐴(𝑉𝑂𝐸)𝐵) ≠ ∅ → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))))
32	25, 30, 31	3syl 19	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝐴(𝑉𝑂𝐸)𝐵) ≠ ∅ → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))))
33	24, 32	pm2.61i 184	. . . . . . . . . . . . . 14 ⊢ ((𝐴(𝑉𝑂𝐸)𝐵) ≠ ∅ → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))))
34	14, 33	syl 18	. . . . . . . . . . . . 13 ⊢ (⟨𝐹, 𝑃⟩ ∈ (𝐴(𝑉𝑂𝐸)𝐵) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))))
35	13, 34	sylbi 220	. . . . . . . . . . . 12 ⊢ (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))))
36	35	pm2.43i 53	. . . . . . . . . . 11 ⊢ (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
37	36	com12 33	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
38	37	anc2ri 566	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))
39		df-3an 1105	. . . . . . . . 9 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) ↔ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))
40	38, 39	imbitrrdi 255	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))
41	12, 40	sylbi 220	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))
42	11, 41	syl 18	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑐 = {⟨𝑓, 𝑝⟩ ∣ 𝜓})} → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))
43		df-mpo 7424	. . . . . . 7 ⊢ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) = {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑐 = {⟨𝑓, 𝑝⟩ ∣ 𝜓})}
44	43	dmeqi 5896	. . . . . 6 ⊢ dom (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) = dom {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑐 = {⟨𝑓, 𝑝⟩ ∣ 𝜓})}
45	42, 44	eleq2s 2883	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))
46	9, 45	biimtrdi 256	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))))
47		3ianor 1124	. . . . 5 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) ↔ (¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V ∨ ¬ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V))
48		df-3or 1104	. . . . . 6 ⊢ ((¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V ∨ ¬ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) ↔ ((¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V) ∨ ¬ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V))
49		ianor 997	. . . . . . . 8 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) ↔ (¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V))
50	25	dmeqd 5897	. . . . . . . . . . 11 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → dom (𝑉𝑂𝐸) = dom ∅)
51	50	eleq2d 2851	. . . . . . . . . 10 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) ↔ ⟨𝐴, 𝐵⟩ ∈ dom ∅))
52		dm0 5912	. . . . . . . . . . 11 ⊢ dom ∅ = ∅
53	52	eleq2i 2857	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom ∅ ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
54	51, 53	bitrdi 290	. . . . . . . . 9 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) ↔ ⟨𝐴, 𝐵⟩ ∈ ∅))
55		noel 4291	. . . . . . . . . 10 ⊢ ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
56	55	pm2.21i 120	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ ∈ ∅ → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))
57	54, 56	biimtrdi 256	. . . . . . . 8 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))))
58	49, 57	sylbir 238	. . . . . . 7 ⊢ ((¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))))
59		anor 998	. . . . . . . . 9 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ↔ ¬ (¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V))
60		id 23	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ V → 𝑉 ∈ V)
61	60	ancri 559	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ V → (𝑉 ∈ V ∧ 𝑉 ∈ V))
62	61	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ∈ V ∧ 𝑉 ∈ V))
63		mpoexga 8080	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ V ∧ 𝑉 ∈ V) → (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V)
64	62, 63	syl 18	. . . . . . . . . 10 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V)
65	64	pm2.24d 152	. . . . . . . . 9 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (¬ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))))
66	59, 65	sylbir 238	. . . . . . . 8 ⊢ (¬ (¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V) → (¬ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))))
67	66	imp 412	. . . . . . 7 ⊢ ((¬ (¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V) ∧ ¬ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))))
68	58, 67	jaoi3 1076	. . . . . 6 ⊢ (((¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V) ∨ ¬ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))))
69	48, 68	sylbi 220	. . . . 5 ⊢ ((¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V ∨ ¬ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))))
70	47, 69	sylbi 220	. . . 4 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜓}) ∈ V) → (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))))
71	46, 70	pm2.61i 184	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom (𝑉𝑂𝐸) → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))
72	1, 71	syl 18	. 2 ⊢ (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))
73	72	pm2.43i 53	1 ⊢ (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  Vcvv 3457  ∅c0 4286  ⟨cop 4597   class class class wbr 5111  {copab 5175   × cxp 5661  dom cdm 5663  ‘cfv 6540  (class class class)co 7419  {coprab 7420   ∈ cmpo 7421

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993

This theorem is used by: (None)