Theorem grumnudlem 44641

Description: Lemma for grumnud 44642. (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 10719	. . . . . . . 8 ⊢ ((𝐺 ∈ Univ ∧ 𝑧 ∈ 𝐺 ∧ 𝑎 ⊆ 𝑧) → 𝑎 ∈ 𝐺)
3	1, 2	syl3an1 1164	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑎 ⊆ 𝑧) → 𝑎 ∈ 𝐺)
4	3	3expia 1122	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → (𝑎 ⊆ 𝑧 → 𝑎 ∈ 𝐺))
5	4	alrimiv 1929	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → ∀𝑎(𝑎 ⊆ 𝑧 → 𝑎 ∈ 𝐺))
6		pwss 4579	. . . . 5 ⊢ (𝒫 𝑧 ⊆ 𝐺 ↔ ∀𝑎(𝑎 ⊆ 𝑧 → 𝑎 ∈ 𝐺))
7	5, 6	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝒫 𝑧 ⊆ 𝐺)
8		ssun1 4132	. . . . . . . . 9 ⊢ 𝒫 𝑧 ⊆ (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))
9		simp3 1139	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) → 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)))
10	8, 9	sseqtrrid 3979	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) → 𝒫 𝑧 ⊆ 𝑤)
11		simp1l3 1270	. . . . . . . . . . 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 1200	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → 𝑖 ∈ 𝑧)
13		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → 𝑗 = 𝑣)
14	13	unieqd 4878	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → ∪ 𝑗 = ∪ 𝑣)
15		simpl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → ℎ = ∪ 𝑣)
16	14, 15	eqtr4d 2775	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → ∪ 𝑗 = ℎ)
17	16	adantll 715	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . 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 1216	. . . . . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . . . 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 1129	. . . . . . . . . . . . . . . . . 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 1194	. . . . . . . . . . . . . . . . . 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 44560	. . . . . . . . . . . . . . . . 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 1268	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → 𝜑)
28	27, 1	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → 𝐺 ∈ Univ)
29		simp2 1138	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → 𝑣 ∈ 𝐺)
30		gruuni 10723	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Univ ∧ 𝑣 ∈ 𝐺) → ∪ 𝑣 ∈ 𝐺)
31	28, 29, 30	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → ∪ 𝑣 ∈ 𝐺)
32	26, 31	rspcime 3583	. . . . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) → 𝜑)
34	33, 1	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) → 𝐺 ∈ Univ)
35		simpl2 1194	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) → 𝑧 ∈ 𝐺)
36		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) → 𝑖 ∈ 𝑧)
37		gruel 10726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Univ ∧ 𝑧 ∈ 𝐺 ∧ 𝑖 ∈ 𝑧) → 𝑖 ∈ 𝐺)
38	34, 35, 36, 37	syl3anc 1374	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) → 𝑖 ∈ 𝐺)
39	38	3ad2ant1 1134	. . . . . . . . . . . . . . . . . 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 581	. . . . . . . . . . . . . . . . 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 3160	. . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → ∃ℎ ∈ 𝐺 𝑖𝐹ℎ)
44		rexex 3068	. . . . . . . . . . . . . . 15 ⊢ (∃ℎ ∈ 𝐺 𝑖𝐹ℎ → ∃ℎ 𝑖𝐹ℎ)
45	43, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → ∃ℎ 𝑖𝐹ℎ)
46	12, 45	cpcoll2d 44615	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → 𝑧 ∈ 𝐺)
49	48	adantr 480	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝐺 ∈ Univ)
51		grumnudlem.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐹 = ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺))
52		inss2 4192	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) ⊆ (𝐺 × 𝐺)
53	51, 52	eqsstri 3982	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐹 ⊆ (𝐺 × 𝐺)
54	53	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝐹 ⊆ (𝐺 × 𝐺))
55		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝑧 ∈ 𝐺)
56	50, 54, 55	grucollcld 44616	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → (𝐹 Coll 𝑧) ∈ 𝐺)
57	27, 49, 56	syl2an2r 686	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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 10726	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Univ ∧ (𝐹 Coll 𝑧) ∈ 𝐺 ∧ ℎ ∈ (𝐹 Coll 𝑧)) → ℎ ∈ 𝐺)
60	47, 57, 58, 59	syl3anc 1374	. . . . . . . . . . . . . . 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 686	. . . . . . . . . . . . . 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 3160	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 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 3265	. . . . . . . . . . . . 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 3131	. . . . . . . . . . . . . 14 ⊢ (∃ℎ ∈ (𝐹 Coll 𝑧)(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)))
67	66	exlimiv 1932	. . . . . . . . . . . . 13 ⊢ (∃𝑗∃ℎ ∈ (𝐹 Coll 𝑧)(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)))
68	64, 67	sylbi 217	. . . . . . . . . . . 12 ⊢ (∃ℎ ∈ (𝐹 Coll 𝑧)∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)))
69	63, 68	syl 17	. . . . . . . . . . 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 4896	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑢 ∈ (𝐹 Coll 𝑧) → ∪ 𝑢 ⊆ ∪ (𝐹 Coll 𝑧))
71		ssun2 4133	. . . . . . . . . . . . . . . . 17 ⊢ ∪ (𝐹 Coll 𝑧) ⊆ (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))
72	70, 71	sstrdi 3948	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑢 ∈ (𝐹 Coll 𝑧) → ∪ 𝑢 ⊆ (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)))
73	72	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)) → ∪ 𝑢 ⊆ (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)))
74		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)) → 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)))
75	73, 74	sseqtrrd 3973	. . . . . . . . . . . . . 14 ⊢ ((𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)) → ∪ 𝑢 ⊆ 𝑤)
76	75	ex 412	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) → (∪ 𝑢 ∈ (𝐹 Coll 𝑧) → ∪ 𝑢 ⊆ 𝑤))
77	76	anim2d 613	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) → ((𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)) → (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
78	77	reximdv 3153	. . . . . . . . . . 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 65	. . . . . . . . . 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 3143	. . . . . . . . 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 3130	. . . . . . . 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 511	. . . . . . 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 1119	. . . . . 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 10718	. . . . . . . 8 ⊢ ((𝐺 ∈ Univ ∧ 𝑧 ∈ 𝐺) → 𝒫 𝑧 ∈ 𝐺)
85	1, 84	sylan 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝒫 𝑧 ∈ 𝐺)
86		gruuni 10723	. . . . . . . 8 ⊢ ((𝐺 ∈ Univ ∧ (𝐹 Coll 𝑧) ∈ 𝐺) → ∪ (𝐹 Coll 𝑧) ∈ 𝐺)
87	1, 56, 86	syl2an2r 686	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → ∪ (𝐹 Coll 𝑧) ∈ 𝐺)
88		gruun 10729	. . . . . . 7 ⊢ ((𝐺 ∈ Univ ∧ 𝒫 𝑧 ∈ 𝐺 ∧ ∪ (𝐹 Coll 𝑧) ∈ 𝐺) → (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∈ 𝐺)
89	50, 85, 87, 88	syl3anc 1374	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∈ 𝐺)
90	83, 89	rspcime 3583	. . . . 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 1929	. . . 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 511	. . 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 3130	. 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 44617	. . 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 17	. 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 257	1 ⊢ (𝜑 → 𝐺 ∈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556  ∪ cuni 4865   class class class wbr 5100  {copab 5162   × cxp 5630  Univcgru 10713   Coll ccoll 44606

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-reg 9509  ax-inf2 9562  ax-ac2 10385

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-tc 9656  df-r1 9688  df-rank 9689  df-card 9863  df-cf 9865  df-acn 9866  df-ac 10038  df-wina 10607  df-ina 10608  df-gru 10714  df-scott 44592  df-coll 44607

This theorem is referenced by:  grumnud  44642