Theorem grumnudlem 44309

Description: Lemma for grumnud 44310. (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 10810	. . . . . . . 8 ⊢ ((𝐺 ∈ Univ ∧ 𝑧 ∈ 𝐺 ∧ 𝑎 ⊆ 𝑧) → 𝑎 ∈ 𝐺)
3	1, 2	syl3an1 1163	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑎 ⊆ 𝑧) → 𝑎 ∈ 𝐺)
4	3	3expia 1121	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → (𝑎 ⊆ 𝑧 → 𝑎 ∈ 𝐺))
5	4	alrimiv 1927	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → ∀𝑎(𝑎 ⊆ 𝑧 → 𝑎 ∈ 𝐺))
6		pwss 4598	. . . . 5 ⊢ (𝒫 𝑧 ⊆ 𝐺 ↔ ∀𝑎(𝑎 ⊆ 𝑧 → 𝑎 ∈ 𝐺))
7	5, 6	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝒫 𝑧 ⊆ 𝐺)
8		ssun1 4153	. . . . . . . . 9 ⊢ 𝒫 𝑧 ⊆ (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))
9		simp3 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) → 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)))
10	8, 9	sseqtrrid 4002	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) → 𝒫 𝑧 ⊆ 𝑤)
11		simp1l3 1269	. . . . . . . . . . 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 1199	. . . . . . . . . . . . . 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 4896	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → ∪ 𝑗 = ∪ 𝑣)
15		simpl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → ℎ = ∪ 𝑣)
16	14, 15	eqtr4d 2773	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → ∪ 𝑗 = ℎ)
17	16	adantll 714	. . . . . . . . . . . . . . . . . . 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 1215	. . . . . . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . . . 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 44228	. . . . . . . . . . . . . . . . 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 1267	. . . . . . . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → 𝑣 ∈ 𝐺)
30		gruuni 10814	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Univ ∧ 𝑣 ∈ 𝐺) → ∪ 𝑣 ∈ 𝐺)
31	28, 29, 30	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → ∪ 𝑣 ∈ 𝐺)
32	26, 31	rspcime 3606	. . . . . . . . . . . . . . . 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 1192	. . . . . . . . . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . . . . . 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 10817	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Univ ∧ 𝑧 ∈ 𝐺 ∧ 𝑖 ∈ 𝑧) → 𝑖 ∈ 𝐺)
38	34, 35, 36, 37	syl3anc 1373	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) → 𝑖 ∈ 𝐺)
39	38	3ad2ant1 1133	. . . . . . . . . . . . . . . . . 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 580	. . . . . . . . . . . . . . . . 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 3162	. . . . . . . . . . . . . . . 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 3066	. . . . . . . . . . . . . . 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 44283	. . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . . . 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 4213	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) ⊆ (𝐺 × 𝐺)
53	51, 52	eqsstri 4005	. . . . . . . . . . . . . . . . . . 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 44284	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → (𝐹 Coll 𝑧) ∈ 𝐺)
57	27, 49, 56	syl2an2r 685	. . . . . . . . . . . . . . . 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 10817	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Univ ∧ (𝐹 Coll 𝑧) ∈ 𝐺 ∧ ℎ ∈ (𝐹 Coll 𝑧)) → ℎ ∈ 𝐺)
60	47, 57, 58, 59	syl3anc 1373	. . . . . . . . . . . . . . 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 685	. . . . . . . . . . . . . 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 3162	. . . . . . . . . . . . 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 3269	. . . . . . . . . . . . 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 3133	. . . . . . . . . . . . . 14 ⊢ (∃ℎ ∈ (𝐹 Coll 𝑧)(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)))
67	66	exlimiv 1930	. . . . . . . . . . . . 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 4913	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑢 ∈ (𝐹 Coll 𝑧) → ∪ 𝑢 ⊆ ∪ (𝐹 Coll 𝑧))
71		ssun2 4154	. . . . . . . . . . . . . . . . 17 ⊢ ∪ (𝐹 Coll 𝑧) ⊆ (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))
72	70, 71	sstrdi 3971	. . . . . . . . . . . . . . . 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 3996	. . . . . . . . . . . . . 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 612	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) → ((𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)) → (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
78	77	reximdv 3155	. . . . . . . . . . 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 3145	. . . . . . . . 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 3132	. . . . . . . 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 1118	. . . . . 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 10809	. . . . . . . 8 ⊢ ((𝐺 ∈ Univ ∧ 𝑧 ∈ 𝐺) → 𝒫 𝑧 ∈ 𝐺)
85	1, 84	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝒫 𝑧 ∈ 𝐺)
86		gruuni 10814	. . . . . . . 8 ⊢ ((𝐺 ∈ Univ ∧ (𝐹 Coll 𝑧) ∈ 𝐺) → ∪ (𝐹 Coll 𝑧) ∈ 𝐺)
87	1, 56, 86	syl2an2r 685	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → ∪ (𝐹 Coll 𝑧) ∈ 𝐺)
88		gruun 10820	. . . . . . 7 ⊢ ((𝐺 ∈ Univ ∧ 𝒫 𝑧 ∈ 𝐺 ∧ ∪ (𝐹 Coll 𝑧) ∈ 𝐺) → (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∈ 𝐺)
89	50, 85, 87, 88	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∈ 𝐺)
90	83, 89	rspcime 3606	. . . . 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 1927	. . . 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 3132	. 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 44285	. . 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 1086  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2713  ∀wral 3051  ∃wrex 3060   ∪ cun 3924   ∩ cin 3925   ⊆ wss 3926  𝒫 cpw 4575  ∪ cuni 4883   class class class wbr 5119  {copab 5181   × cxp 5652  Univcgru 10804   Coll ccoll 44274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-reg 9606  ax-inf2 9655  ax-ac2 10477

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-tc 9751  df-r1 9778  df-rank 9779  df-card 9953  df-cf 9955  df-acn 9956  df-ac 10130  df-wina 10698  df-ina 10699  df-gru 10805  df-scott 44260  df-coll 44275

This theorem is referenced by:  grumnud  44310