Theorem grumnudlem 45530

Description: Lemma for grumnud 45531. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypotheses

Ref	Expression
grumnudlem.1	⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))}
grumnudlem.2	⊢ (𝜑 → 𝐺 ∈ Univ)
grumnudlem.3	⊢ 𝐹 = ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺))
grumnudlem.4	⊢ ((𝑖 ∈ 𝐺 ∧ ℎ ∈ 𝐺) → (𝑖𝐹ℎ ↔ ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
grumnudlem.5	⊢ ((ℎ ∈ (𝐹 Coll 𝑧) ∧ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)))

Assertion

Ref	Expression
grumnudlem	⊢ (𝜑 → 𝐺 ∈ 𝑀)

Distinct variable groups:   𝜑,𝑧,𝑓,ℎ,𝑗   𝑧,𝐺   𝑓,ℎ,𝑗,𝐺   𝜑,𝑓,ℎ,𝑖,𝑗   𝑢,ℎ,𝑖,𝑗,𝐹   𝑧,𝑖,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙,𝑓,𝐺   𝑧,𝑢,𝑟,𝑓,𝑘,𝑚,𝑛,𝐺,𝑝,𝑙,𝑖

Allowed substitution hints:   𝜑(𝑢, 𝑘, 𝑚, 𝑛, 𝑟, 𝑞, 𝑝, 𝑏, 𝑐, 𝑑, 𝑙)   𝐹(𝑧, 𝑓, 𝑘, 𝑚, 𝑛, 𝑟, 𝑞, 𝑝, 𝑏, 𝑐, 𝑑, 𝑙)   𝐺(𝑏, 𝑐, 𝑑)   𝑀(𝑧, 𝑢, 𝑓, ℎ, 𝑖, 𝑗, 𝑘, 𝑚, 𝑛, 𝑟, 𝑞, 𝑝, 𝑏, 𝑐, 𝑑, 𝑙)

Proof of Theorem grumnudlem

Dummy variables 𝑎 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grumnudlem.2	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Univ)
2		gruss 10920	. . . . . . . 8 ⊢ ((𝐺 ∈ Univ ∧ 𝑧 ∈ 𝐺 ∧ 𝑎 ⊆ 𝑧) → 𝑎 ∈ 𝐺)
3	1, 2	syl3an1 1181	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑎 ⊆ 𝑧) → 𝑎 ∈ 𝐺)
4	3	3expia 1139	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → (𝑎 ⊆ 𝑧 → 𝑎 ∈ 𝐺))
5	4	alrimiv 1962	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → ∀𝑎(𝑎 ⊆ 𝑧 → 𝑎 ∈ 𝐺))
6		pwss 4585	. . . . 5 ⊢ (𝒫 𝑧 ⊆ 𝐺 ↔ ∀𝑎(𝑎 ⊆ 𝑧 → 𝑎 ∈ 𝐺))
7	5, 6	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝒫 𝑧 ⊆ 𝐺)
8		ssun1 4179	. . . . . . . . 9 ⊢ 𝒫 𝑧 ⊆ (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))
9		simp3 1156	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) → 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)))
10	8, 9	sseqtrrid 3956	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) → 𝒫 𝑧 ⊆ 𝑤)
11		simp1l3 1287	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)))
12		simp1r 1217	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → 𝑖 ∈ 𝑧)
13		simpr 490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → 𝑗 = 𝑣)
14	13	unieqd 4885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → ∪ 𝑗 = ∪ 𝑣)
15		simpl 488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → ℎ = ∪ 𝑣)
16	14, 15	eqtr4d 2801	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → ∪ 𝑗 = ℎ)
17	16	adantll 727	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ = ∪ 𝑣) ∧ 𝑗 = 𝑣) → ∪ 𝑗 = ℎ)
18		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ = ∪ 𝑣) ∧ 𝑗 = 𝑣) → 𝑗 = 𝑣)
19		simpll3 1233	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ = ∪ 𝑣) ∧ 𝑗 = 𝑣) → (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓))
20	19	simprd 501	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ = ∪ 𝑣) ∧ 𝑗 = 𝑣) → 𝑣 ∈ 𝑓)
21	18, 20	eqeltrd 2864	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ = ∪ 𝑣) ∧ 𝑗 = 𝑣) → 𝑗 ∈ 𝑓)
22	19	simpld 500	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ = ∪ 𝑣) ∧ 𝑗 = 𝑣) → 𝑖 ∈ 𝑣)
23	22, 18	eleqtrrd 2867	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ = ∪ 𝑣) ∧ 𝑗 = 𝑣) → 𝑖 ∈ 𝑗)
24	17, 21, 23	3jca 1146	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ = ∪ 𝑣) ∧ 𝑗 = 𝑣) → (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗))
25		simpl2 1211	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ = ∪ 𝑣) → 𝑣 ∈ 𝐺)
26	24, 25	rr-spce 45463	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ = ∪ 𝑣) → ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗))
27		simp1l1 1285	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → 𝜑)
28	27, 1	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → 𝐺 ∈ Univ)
29		simp2 1155	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → 𝑣 ∈ 𝐺)
30		gruuni 10924	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Univ ∧ 𝑣 ∈ 𝐺) → ∪ 𝑣 ∈ 𝐺)
31	28, 29, 30	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → ∪ 𝑣 ∈ 𝐺)
32	26, 31	rspcime 3585	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → ∃ℎ ∈ 𝐺 ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗))
33		simpl1 1210	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) → 𝜑)
34	33, 1	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) → 𝐺 ∈ Univ)
35		simpl2 1211	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) → 𝑧 ∈ 𝐺)
36		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) → 𝑖 ∈ 𝑧)
37		gruel 10927	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Univ ∧ 𝑧 ∈ 𝐺 ∧ 𝑖 ∈ 𝑧) → 𝑖 ∈ 𝐺)
38	34, 35, 36, 37	syl3anc 1399	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) → 𝑖 ∈ 𝐺)
39	38	3ad2ant1 1151	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → 𝑖 ∈ 𝐺)
40		grumnudlem.4	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝐺 ∧ ℎ ∈ 𝐺) → (𝑖𝐹ℎ ↔ ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
41	39, 40	sylan 592	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ ∈ 𝐺) → (𝑖𝐹ℎ ↔ ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
42	41	rexbidva 3188	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → (∃ℎ ∈ 𝐺 𝑖𝐹ℎ ↔ ∃ℎ ∈ 𝐺 ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
43	32, 42	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → ∃ℎ ∈ 𝐺 𝑖𝐹ℎ)
44		rexex 3096	. . . . . . . . . . . . . . 15 ⊢ (∃ℎ ∈ 𝐺 𝑖𝐹ℎ → ∃ℎ 𝑖𝐹ℎ)
45	43, 44	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → ∃ℎ 𝑖𝐹ℎ)
46	12, 45	cpcoll2d 45504	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → ∃ℎ ∈ (𝐹 Coll 𝑧)𝑖𝐹ℎ)
47	28	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ ∈ (𝐹 Coll 𝑧)) → 𝐺 ∈ Univ)
48	35	3ad2ant1 1151	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → 𝑧 ∈ 𝐺)
49	48	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ ∈ (𝐹 Coll 𝑧)) → 𝑧 ∈ 𝐺)
50	1	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝐺 ∈ Univ)
51		grumnudlem.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐹 = ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺))
52		inss2 4137	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) ⊆ (𝐺 × 𝐺)
53	51, 52	eqsstri 3959	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐹 ⊆ (𝐺 × 𝐺)
54	53	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝐹 ⊆ (𝐺 × 𝐺))
55		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝑧 ∈ 𝐺)
56	50, 54, 55	grucollcld 45505	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → (𝐹 Coll 𝑧) ∈ 𝐺)
57	27, 49, 56	syl2an2r 698	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ ∈ (𝐹 Coll 𝑧)) → (𝐹 Coll 𝑧) ∈ 𝐺)
58		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ ∈ (𝐹 Coll 𝑧)) → ℎ ∈ (𝐹 Coll 𝑧))
59		gruel 10927	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Univ ∧ (𝐹 Coll 𝑧) ∈ 𝐺 ∧ ℎ ∈ (𝐹 Coll 𝑧)) → ℎ ∈ 𝐺)
60	47, 57, 58, 59	syl3anc 1399	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ ∈ (𝐹 Coll 𝑧)) → ℎ ∈ 𝐺)
61	39, 60, 40	syl2an2r 698	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) ∧ ℎ ∈ (𝐹 Coll 𝑧)) → (𝑖𝐹ℎ ↔ ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
62	61	rexbidva 3188	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → (∃ℎ ∈ (𝐹 Coll 𝑧)𝑖𝐹ℎ ↔ ∃ℎ ∈ (𝐹 Coll 𝑧)∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
63	46, 62	mpbid 235	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → ∃ℎ ∈ (𝐹 Coll 𝑧)∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗))
64		rexcom4 3293	. . . . . . . . . . . . 13 ⊢ (∃ℎ ∈ (𝐹 Coll 𝑧)∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗) ↔ ∃𝑗∃ℎ ∈ (𝐹 Coll 𝑧)(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗))
65		grumnudlem.5	. . . . . . . . . . . . . . 15 ⊢ ((ℎ ∈ (𝐹 Coll 𝑧) ∧ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)))
66	65	rexlimiva 3159	. . . . . . . . . . . . . 14 ⊢ (∃ℎ ∈ (𝐹 Coll 𝑧)(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)))
67	66	exlimiv 1965	. . . . . . . . . . . . 13 ⊢ (∃𝑗∃ℎ ∈ (𝐹 Coll 𝑧)(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)))
68	64, 67	sylbi 220	. . . . . . . . . . . 12 ⊢ (∃ℎ ∈ (𝐹 Coll 𝑧)∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)))
69	63, 68	syl 18	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)))
70		elssuni 4904	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑢 ∈ (𝐹 Coll 𝑧) → ∪ 𝑢 ⊆ ∪ (𝐹 Coll 𝑧))
71		ssun2 4180	. . . . . . . . . . . . . . . . 17 ⊢ ∪ (𝐹 Coll 𝑧) ⊆ (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))
72	70, 71	sstrdi 3925	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑢 ∈ (𝐹 Coll 𝑧) → ∪ 𝑢 ⊆ (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)))
73	72	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)) → ∪ 𝑢 ⊆ (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)))
74		simpl 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)) → 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)))
75	73, 74	sseqtrrd 3950	. . . . . . . . . . . . . 14 ⊢ ((𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)) → ∪ 𝑢 ⊆ 𝑤)
76	75	ex 418	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) → (∪ 𝑢 ∈ (𝐹 Coll 𝑧) → ∪ 𝑢 ⊆ 𝑤))
77	76	anim2d 624	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) → ((𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)) → (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
78	77	reximdv 3181	. . . . . . . . . . 11 ⊢ (𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) → (∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
79	11, 69, 78	sylc 66	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))
80	79	rexlimdv3a 3171	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) → (∃𝑣 ∈ 𝐺 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
81	80	ralrimiva 3158	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) → ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝐺 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
82	10, 81	jca 521	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) → (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝐺 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
83	82	3expa 1136	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺) ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) → (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝐺 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
84		grupw 10919	. . . . . . . 8 ⊢ ((𝐺 ∈ Univ ∧ 𝑧 ∈ 𝐺) → 𝒫 𝑧 ∈ 𝐺)
85	1, 84	sylan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝒫 𝑧 ∈ 𝐺)
86		gruuni 10924	. . . . . . . 8 ⊢ ((𝐺 ∈ Univ ∧ (𝐹 Coll 𝑧) ∈ 𝐺) → ∪ (𝐹 Coll 𝑧) ∈ 𝐺)
87	1, 56, 86	syl2an2r 698	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → ∪ (𝐹 Coll 𝑧) ∈ 𝐺)
88		gruun 10930	. . . . . . 7 ⊢ ((𝐺 ∈ Univ ∧ 𝒫 𝑧 ∈ 𝐺 ∧ ∪ (𝐹 Coll 𝑧) ∈ 𝐺) → (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∈ 𝐺)
89	50, 85, 87, 88	syl3anc 1399	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∈ 𝐺)
90	83, 89	rspcime 3585	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → ∃𝑤 ∈ 𝐺 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝐺 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
91	90	alrimiv 1962	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → ∀𝑓∃𝑤 ∈ 𝐺 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝐺 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
92	7, 91	jca 521	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → (𝒫 𝑧 ⊆ 𝐺 ∧ ∀𝑓∃𝑤 ∈ 𝐺 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝐺 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
93	92	ralrimiva 3158	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐺 (𝒫 𝑧 ⊆ 𝐺 ∧ ∀𝑓∃𝑤 ∈ 𝐺 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝐺 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
94		grumnudlem.1	. . . 4 ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))}
95	94	ismnu 45506	. . 3 ⊢ (𝐺 ∈ Univ → (𝐺 ∈ 𝑀 ↔ ∀𝑧 ∈ 𝐺 (𝒫 𝑧 ⊆ 𝐺 ∧ ∀𝑓∃𝑤 ∈ 𝐺 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝐺 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))))
96	1, 95	syl 18	. 2 ⊢ (𝜑 → (𝐺 ∈ 𝑀 ↔ ∀𝑧 ∈ 𝐺 (𝒫 𝑧 ⊆ 𝐺 ∧ ∀𝑓∃𝑤 ∈ 𝐺 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝐺 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))))
97	93, 96	mpbird 260	1 ⊢ (𝜑 → 𝐺 ∈ 𝑀)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092   ∪ cun 4152  𝒫 cpw 4560  ∪ cuni 4872   class class class wbr 5108  {copab 5173   × cxp 5656  Univcgru 10914   Coll ccoll 45495

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-reg 9623  ax-inf2 9679  ax-ac2 10580

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-tc 9773  df-r1 9805  df-rank 9806  df-scott 9966  df-card 10057  df-cf 10059  df-acn 10060  df-ac 10234  df-wina 10808  df-ina 10809  df-gru 10915  df-coll 45496

This theorem is used by:  grumnud  45531