Theorem efgval 18130

Description: Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)

Hypotheses

Ref	Expression
efgval.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
efgval.r	⊢ ∼ = ( ~FG ‘𝐼)

Assertion

Ref	Expression
efgval	⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))}

Distinct variable groups:   𝑦,𝑟,𝑧,𝑛,𝑥,𝑊   ∼ ,𝑟,𝑥,𝑦,𝑧   𝑛,𝐼,𝑟,𝑥,𝑦,𝑧

Allowed substitution hint:   ∼ (𝑛)

Proof of Theorem efgval

Dummy variables 𝑖 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.r	. 2 ⊢ ∼ = ( ~FG ‘𝐼)
2		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑖 ∈ V
3		2on 7568	. . . . . . . . . . . . 13 ⊢ 2𝑜 ∈ On
4	3	elexi 3213	. . . . . . . . . . . 12 ⊢ 2𝑜 ∈ V
5	2, 4	xpex 6962	. . . . . . . . . . 11 ⊢ (𝑖 × 2𝑜) ∈ V
6		wrdexg 13315	. . . . . . . . . . 11 ⊢ ((𝑖 × 2𝑜) ∈ V → Word (𝑖 × 2𝑜) ∈ V)
7		fvi 6255	. . . . . . . . . . 11 ⊢ (Word (𝑖 × 2𝑜) ∈ V → ( I ‘Word (𝑖 × 2𝑜)) = Word (𝑖 × 2𝑜))
8	5, 6, 7	mp2b 10	. . . . . . . . . 10 ⊢ ( I ‘Word (𝑖 × 2𝑜)) = Word (𝑖 × 2𝑜)
9		xpeq1 5128	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (𝑖 × 2𝑜) = (𝐼 × 2𝑜))
10		wrdeq 13327	. . . . . . . . . . . 12 ⊢ ((𝑖 × 2𝑜) = (𝐼 × 2𝑜) → Word (𝑖 × 2𝑜) = Word (𝐼 × 2𝑜))
11	9, 10	syl 17	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → Word (𝑖 × 2𝑜) = Word (𝐼 × 2𝑜))
12	11	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ( I ‘Word (𝑖 × 2𝑜)) = ( I ‘Word (𝐼 × 2𝑜)))
13	8, 12	syl5eqr 2670	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → Word (𝑖 × 2𝑜) = ( I ‘Word (𝐼 × 2𝑜)))
14		efgval.w	. . . . . . . . 9 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
15	13, 14	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → Word (𝑖 × 2𝑜) = 𝑊)
16		ereq2 7750	. . . . . . . 8 ⊢ (Word (𝑖 × 2𝑜) = 𝑊 → (𝑟 Er Word (𝑖 × 2𝑜) ↔ 𝑟 Er 𝑊))
17	15, 16	syl 17	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑟 Er Word (𝑖 × 2𝑜) ↔ 𝑟 Er 𝑊))
18		raleq 3138	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (∀𝑦 ∈ 𝑖 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩) ↔ ∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)))
19	18	ralbidv 2986	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝑖 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩) ↔ ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)))
20	15, 19	raleqbidv 3152	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (∀𝑥 ∈ Word (𝑖 × 2𝑜)∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝑖 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩) ↔ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)))
21	17, 20	anbi12d 747	. . . . . 6 ⊢ (𝑖 = 𝐼 → ((𝑟 Er Word (𝑖 × 2𝑜) ∧ ∀𝑥 ∈ Word (𝑖 × 2𝑜)∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝑖 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)) ↔ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))))
22	21	abbidv 2741	. . . . 5 ⊢ (𝑖 = 𝐼 → {𝑟 ∣ (𝑟 Er Word (𝑖 × 2𝑜) ∧ ∀𝑥 ∈ Word (𝑖 × 2𝑜)∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝑖 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))} = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))})
23	22	inteqd 4480	. . . 4 ⊢ (𝑖 = 𝐼 → ∩ {𝑟 ∣ (𝑟 Er Word (𝑖 × 2𝑜) ∧ ∀𝑥 ∈ Word (𝑖 × 2𝑜)∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝑖 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))} = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))})
24		df-efg 18122	. . . 4 ⊢ ~FG = (𝑖 ∈ V ↦ ∩ {𝑟 ∣ (𝑟 Er Word (𝑖 × 2𝑜) ∧ ∀𝑥 ∈ Word (𝑖 × 2𝑜)∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝑖 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))})
25	14	efglem 18129	. . . . 5 ⊢ ∃𝑟(𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))
26		intexab 4822	. . . . 5 ⊢ (∃𝑟(𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)) ↔ ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))} ∈ V)
27	25, 26	mpbi 220	. . . 4 ⊢ ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))} ∈ V
28	23, 24, 27	fvmpt 6282	. . 3 ⊢ (𝐼 ∈ V → ( ~FG ‘𝐼) = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))})
29		fvprc 6185	. . . 4 ⊢ (¬ 𝐼 ∈ V → ( ~FG ‘𝐼) = ∅)
30		abn0 3954	. . . . . . . 8 ⊢ ({𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))} ≠ ∅ ↔ ∃𝑟(𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)))
31	25, 30	mpbir 221	. . . . . . 7 ⊢ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))} ≠ ∅
32		intssuni 4499	. . . . . . 7 ⊢ ({𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))} ≠ ∅ → ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))} ⊆ ∪ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))})
33	31, 32	ax-mp 5	. . . . . 6 ⊢ ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))} ⊆ ∪ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))}
34		erssxp 7765	. . . . . . . . . . . 12 ⊢ (𝑟 Er 𝑊 → 𝑟 ⊆ (𝑊 × 𝑊))
35	14	efgrcl 18128	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
36	35	simpld 475	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑊 → 𝐼 ∈ V)
37	36	con3i 150	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝐼 ∈ V → ¬ 𝑥 ∈ 𝑊)
38	37	eq0rdv 3979	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐼 ∈ V → 𝑊 = ∅)
39	38	xpeq2d 5139	. . . . . . . . . . . . . 14 ⊢ (¬ 𝐼 ∈ V → (𝑊 × 𝑊) = (𝑊 × ∅))
40		xp0 5552	. . . . . . . . . . . . . 14 ⊢ (𝑊 × ∅) = ∅
41	39, 40	syl6eq 2672	. . . . . . . . . . . . 13 ⊢ (¬ 𝐼 ∈ V → (𝑊 × 𝑊) = ∅)
42		ss0b 3973	. . . . . . . . . . . . 13 ⊢ ((𝑊 × 𝑊) ⊆ ∅ ↔ (𝑊 × 𝑊) = ∅)
43	41, 42	sylibr 224	. . . . . . . . . . . 12 ⊢ (¬ 𝐼 ∈ V → (𝑊 × 𝑊) ⊆ ∅)
44	34, 43	sylan9ssr 3617	. . . . . . . . . . 11 ⊢ ((¬ 𝐼 ∈ V ∧ 𝑟 Er 𝑊) → 𝑟 ⊆ ∅)
45	44	ex 450	. . . . . . . . . 10 ⊢ (¬ 𝐼 ∈ V → (𝑟 Er 𝑊 → 𝑟 ⊆ ∅))
46	45	adantrd 484	. . . . . . . . 9 ⊢ (¬ 𝐼 ∈ V → ((𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)) → 𝑟 ⊆ ∅))
47	46	alrimiv 1855	. . . . . . . 8 ⊢ (¬ 𝐼 ∈ V → ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)) → 𝑟 ⊆ ∅))
48		sseq1 3626	. . . . . . . . 9 ⊢ (𝑤 = 𝑟 → (𝑤 ⊆ ∅ ↔ 𝑟 ⊆ ∅))
49	48	ralab2 3371	. . . . . . . 8 ⊢ (∀𝑤 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))}𝑤 ⊆ ∅ ↔ ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)) → 𝑟 ⊆ ∅))
50	47, 49	sylibr 224	. . . . . . 7 ⊢ (¬ 𝐼 ∈ V → ∀𝑤 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))}𝑤 ⊆ ∅)
51		unissb 4469	. . . . . . 7 ⊢ (∪ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))} ⊆ ∅ ↔ ∀𝑤 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))}𝑤 ⊆ ∅)
52	50, 51	sylibr 224	. . . . . 6 ⊢ (¬ 𝐼 ∈ V → ∪ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))} ⊆ ∅)
53	33, 52	syl5ss 3614	. . . . 5 ⊢ (¬ 𝐼 ∈ V → ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))} ⊆ ∅)
54		ss0 3974	. . . . 5 ⊢ (∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))} ⊆ ∅ → ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))} = ∅)
55	53, 54	syl 17	. . . 4 ⊢ (¬ 𝐼 ∈ V → ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))} = ∅)
56	29, 55	eqtr4d 2659	. . 3 ⊢ (¬ 𝐼 ∈ V → ( ~FG ‘𝐼) = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))})
57	28, 56	pm2.61i 176	. 2 ⊢ ( ~FG ‘𝐼) = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))}
58	1, 57	eqtri 2644	1 ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ⟨cop 4183  ⟨cotp 4185  ∪ cuni 4436  ∩ cint 4475   class class class wbr 4653   I cid 5023   × cxp 5112  Oncon0 5723  ‘cfv 5888  (class class class)co 6650  1_𝑜c1o 7553  2_𝑜c2o 7554   Er wer 7739  0cc0 9936  ...cfz 12326  #chash 13117  Word cword 13291   splice csplice 13296  ⟨“cs2 13586   ~_FG cefg 18119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-s2 13593  df-efg 18122

This theorem is referenced by:  efger  18131  efgi  18132  efgval2  18137  frgpuplem  18185