Theorem grumnudlem 44281

Description: Lemma for grumnud 44282. (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 10756	. . . . . . . 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 4589	. . . . 5 ⊢ (𝒫 𝑧 ⊆ 𝐺 ↔ ∀𝑎(𝑎 ⊆ 𝑧 → 𝑎 ∈ 𝐺))
7	5, 6	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝒫 𝑧 ⊆ 𝐺)
8		ssun1 4144	. . . . . . . . 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 3993	. . . . . . . 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 4887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → ∪ 𝑗 = ∪ 𝑣)
15		simpl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → ℎ = ∪ 𝑣)
16	14, 15	eqtr4d 2768	. . . . . . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . . . . 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 2832	. . . . . . . . . . . . . . . . . . 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 44200	. . . . . . . . . . . . . . . . 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 10760	. . . . . . . . . . . . . . . . . 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 3596	. . . . . . . . . . . . . . . 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 10763	. . . . . . . . . . . . . . . . . . . 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 3156	. . . . . . . . . . . . . . . 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 3060	. . . . . . . . . . . . . . 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 44255	. . . . . . . . . . . . 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 4204	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) ⊆ (𝐺 × 𝐺)
53	51, 52	eqsstri 3996	. . . . . . . . . . . . . . . . . . 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 44256	. . . . . . . . . . . . . . . . 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 10763	. . . . . . . . . . . . . . . 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 3156	. . . . . . . . . . . . 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 3127	. . . . . . . . . . . . . 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 4904	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑢 ∈ (𝐹 Coll 𝑧) → ∪ 𝑢 ⊆ ∪ (𝐹 Coll 𝑧))
71		ssun2 4145	. . . . . . . . . . . . . . . . 17 ⊢ ∪ (𝐹 Coll 𝑧) ⊆ (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))
72	70, 71	sstrdi 3962	. . . . . . . . . . . . . . . 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 3987	. . . . . . . . . . . . . 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 3149	. . . . . . . . . . 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 3139	. . . . . . . . 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 3126	. . . . . . . 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 10755	. . . . . . . 8 ⊢ ((𝐺 ∈ Univ ∧ 𝑧 ∈ 𝐺) → 𝒫 𝑧 ∈ 𝐺)
85	1, 84	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝒫 𝑧 ∈ 𝐺)
86		gruuni 10760	. . . . . . . 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 10766	. . . . . . 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 3596	. . . . 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 3126	. 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 44257	. . 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 2109  {cab 2708  ∀wral 3045  ∃wrex 3054   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  𝒫 cpw 4566  ∪ cuni 4874   class class class wbr 5110  {copab 5172   × cxp 5639  Univcgru 10750   Coll ccoll 44246

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-reg 9552  ax-inf2 9601  ax-ac2 10423

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-tc 9697  df-r1 9724  df-rank 9725  df-card 9899  df-cf 9901  df-acn 9902  df-ac 10076  df-wina 10644  df-ina 10645  df-gru 10751  df-scott 44232  df-coll 44247

This theorem is referenced by:  grumnud  44282