grumnudlem - Mathbox for Rohan Ridenour

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Theorem grumnudlem 43346

Description: Lemma for grumnud 43347. (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 10793	. . . . . . . 8 ⊢ ((𝐺 ∈ Univ ∧ 𝑧 ∈ 𝐺 ∧ 𝑎 ⊆ 𝑧) → 𝑎 ∈ 𝐺)
3	1, 2	syl3an1 1161	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑎 ⊆ 𝑧) → 𝑎 ∈ 𝐺)
4	3	3expia 1119	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → (𝑎 ⊆ 𝑧 → 𝑎 ∈ 𝐺))
5	4	alrimiv 1928	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → ∀𝑎(𝑎 ⊆ 𝑧 → 𝑎 ∈ 𝐺))
6		pwss 4624	. . . . 5 ⊢ (𝒫 𝑧 ⊆ 𝐺 ↔ ∀𝑎(𝑎 ⊆ 𝑧 → 𝑎 ∈ 𝐺))
7	5, 6	sylibr 233	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝒫 𝑧 ⊆ 𝐺)
8		ssun1 4171	. . . . . . . . 9 ⊢ 𝒫 𝑧 ⊆ (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))
9		simp3 1136	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) → 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)))
10	8, 9	sseqtrrid 4034	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) → 𝒫 𝑧 ⊆ 𝑤)
11		simp1l3 1266	. . . . . . . . . . 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 1196	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → 𝑖 ∈ 𝑧)
13		simpr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → 𝑗 = 𝑣)
14	13	unieqd 4921	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → ∪ 𝑗 = ∪ 𝑣)
15		simpl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → ℎ = ∪ 𝑣)
16	14, 15	eqtr4d 2773	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ = ∪ 𝑣 ∧ 𝑗 = 𝑣) → ∪ 𝑗 = ℎ)
17	16	adantll 710	. . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . 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 1212	. . . . . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . . . . . 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 1126	. . . . . . . . . . . . . . . . . 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 1190	. . . . . . . . . . . . . . . . . 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 43258	. . . . . . . . . . . . . . . . 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 1264	. . . . . . . . . . . . . . . . . . 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 1135	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → 𝑣 ∈ 𝐺)
30		gruuni 10797	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Univ ∧ 𝑣 ∈ 𝐺) → ∪ 𝑣 ∈ 𝐺)
31	28, 29, 30	syl2anc 582	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → ∪ 𝑣 ∈ 𝐺)
32	26, 31	rspcime 3615	. . . . . . . . . . . . . . . 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 1189	. . . . . . . . . . . . . . . . . . . . 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 1190	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) → 𝑧 ∈ 𝐺)
36		simpr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) → 𝑖 ∈ 𝑧)
37		gruel 10800	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Univ ∧ 𝑧 ∈ 𝐺 ∧ 𝑖 ∈ 𝑧) → 𝑖 ∈ 𝐺)
38	34, 35, 36, 37	syl3anc 1369	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) → 𝑖 ∈ 𝐺)
39	38	3ad2ant1 1131	. . . . . . . . . . . . . . . . . 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 578	. . . . . . . . . . . . . . . . 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 3174	. . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → ∃ℎ ∈ 𝐺 𝑖𝐹ℎ)
44		rexex 3074	. . . . . . . . . . . . . . 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 43320	. . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐺 ∧ 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))) ∧ 𝑖 ∈ 𝑧) ∧ 𝑣 ∈ 𝐺 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)) → 𝑧 ∈ 𝐺)
49	48	adantr 479	. . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝐺 ∈ Univ)
51		grumnudlem.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐹 = ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺))
52		inss2 4228	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) ⊆ (𝐺 × 𝐺)
53	51, 52	eqsstri 4015	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐹 ⊆ (𝐺 × 𝐺)
54	53	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝐹 ⊆ (𝐺 × 𝐺))
55		simpr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝑧 ∈ 𝐺)
56	50, 54, 55	grucollcld 43321	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → (𝐹 Coll 𝑧) ∈ 𝐺)
57	27, 49, 56	syl2an2r 681	. . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . 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 10800	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Univ ∧ (𝐹 Coll 𝑧) ∈ 𝐺 ∧ ℎ ∈ (𝐹 Coll 𝑧)) → ℎ ∈ 𝐺)
60	47, 57, 58, 59	syl3anc 1369	. . . . . . . . . . . . . . 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 681	. . . . . . . . . . . . . 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 3174	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 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 3283	. . . . . . . . . . . . 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 3145	. . . . . . . . . . . . . 14 ⊢ (∃ℎ ∈ (𝐹 Coll 𝑧)(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)))
67	66	exlimiv 1931	. . . . . . . . . . . . 13 ⊢ (∃𝑗∃ℎ ∈ (𝐹 Coll 𝑧)(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)))
68	64, 67	sylbi 216	. . . . . . . . . . . 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 4940	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑢 ∈ (𝐹 Coll 𝑧) → ∪ 𝑢 ⊆ ∪ (𝐹 Coll 𝑧))
71		ssun2 4172	. . . . . . . . . . . . . . . . 17 ⊢ ∪ (𝐹 Coll 𝑧) ⊆ (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧))
72	70, 71	sstrdi 3993	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑢 ∈ (𝐹 Coll 𝑧) → ∪ 𝑢 ⊆ (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)))
73	72	adantl 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)) → ∪ 𝑢 ⊆ (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)))
74		simpl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)) → 𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)))
75	73, 74	sseqtrrd 4022	. . . . . . . . . . . . . 14 ⊢ ((𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)) → ∪ 𝑢 ⊆ 𝑤)
76	75	ex 411	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) → (∪ 𝑢 ∈ (𝐹 Coll 𝑧) → ∪ 𝑢 ⊆ 𝑤))
77	76	anim2d 610	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) → ((𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)) → (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
78	77	reximdv 3168	. . . . . . . . . . 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 3157	. . . . . . . . 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 3144	. . . . . . . 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 510	. . . . . . 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 1116	. . . . . 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 10792	. . . . . . . 8 ⊢ ((𝐺 ∈ Univ ∧ 𝑧 ∈ 𝐺) → 𝒫 𝑧 ∈ 𝐺)
85	1, 84	sylan 578	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → 𝒫 𝑧 ∈ 𝐺)
86		gruuni 10797	. . . . . . . 8 ⊢ ((𝐺 ∈ Univ ∧ (𝐹 Coll 𝑧) ∈ 𝐺) → ∪ (𝐹 Coll 𝑧) ∈ 𝐺)
87	1, 56, 86	syl2an2r 681	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → ∪ (𝐹 Coll 𝑧) ∈ 𝐺)
88		gruun 10803	. . . . . . 7 ⊢ ((𝐺 ∈ Univ ∧ 𝒫 𝑧 ∈ 𝐺 ∧ ∪ (𝐹 Coll 𝑧) ∈ 𝐺) → (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∈ 𝐺)
89	50, 85, 87, 88	syl3anc 1369	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐺) → (𝒫 𝑧 ∪ ∪ (𝐹 Coll 𝑧)) ∈ 𝐺)
90	83, 89	rspcime 3615	. . . . 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 1928	. . . 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 510	. . 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 3144	. 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 43322	. . 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 256	1 ⊢ (𝜑 → 𝐺 ∈ 𝑀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 ∀wal 1537 = wceq 1539 ∃wex 1779 ∈ wcel 2104 {cab 2707 ∀wral 3059 ∃wrex 3068 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 𝒫 cpw 4601 ∪ cuni 4907 class class class wbr 5147 {copab 5209 × cxp 5673 Univcgru 10787 Coll ccoll 43311

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-reg 9589 ax-inf2 9638 ax-ac2 10460

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-tc 9734 df-r1 9761 df-rank 9762 df-card 9936 df-cf 9938 df-acn 9939 df-ac 10113 df-wina 10681 df-ina 10682 df-gru 10788 df-scott 43297 df-coll 43312

This theorem is referenced by: grumnud 43347

W3C validator