Theorem efgval 19759

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 3492	. . . . . . . . . . . 12 ⊢ 𝑖 ∈ V
3		2on 8536	. . . . . . . . . . . . 13 ⊢ 2o ∈ On
4	3	elexi 3511	. . . . . . . . . . . 12 ⊢ 2o ∈ V
5	2, 4	xpex 7788	. . . . . . . . . . 11 ⊢ (𝑖 × 2o) ∈ V
6		wrdexg 14572	. . . . . . . . . . 11 ⊢ ((𝑖 × 2o) ∈ V → Word (𝑖 × 2o) ∈ V)
7		fvi 6998	. . . . . . . . . . 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 5714	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (𝑖 × 2o) = (𝐼 × 2o))
10		wrdeq 14584	. . . . . . . . . . . 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 6924	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ( I ‘Word (𝑖 × 2o)) = ( I ‘Word (𝐼 × 2o)))
13	8, 12	eqtr3id 2794	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → Word (𝑖 × 2o) = ( I ‘Word (𝐼 × 2o)))
14		efgval.w	. . . . . . . . 9 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
15	13, 14	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → Word (𝑖 × 2o) = 𝑊)
16		ereq2 8771	. . . . . . . 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 3331	. . . . . . . . 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 3184	. . . . . . . 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 3354	. . . . . . 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 631	. . . . . 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 2811	. . . . 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 4975	. . . 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 19751	. . . 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 19758	. . . . 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 5364	. . . . 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 230	. . . 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 7029	. . 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 6912	. . . 4 ⊢ (¬ 𝐼 ∈ V → ( ~FG ‘𝐼) = ∅)
30		abn0 4408	. . . . . . . 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 231	. . . . . . 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 4994	. . . . . . 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 8786	. . . . . . . . . . . 12 ⊢ (𝑟 Er 𝑊 → 𝑟 ⊆ (𝑊 × 𝑊))
35	14	efgrcl 19757	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
36	35	simpld 494	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑊 → 𝐼 ∈ V)
37	36	con3i 154	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝐼 ∈ V → ¬ 𝑥 ∈ 𝑊)
38	37	eq0rdv 4430	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐼 ∈ V → 𝑊 = ∅)
39	38	xpeq2d 5730	. . . . . . . . . . . . . 14 ⊢ (¬ 𝐼 ∈ V → (𝑊 × 𝑊) = (𝑊 × ∅))
40		xp0 6189	. . . . . . . . . . . . . 14 ⊢ (𝑊 × ∅) = ∅
41	39, 40	eqtrdi 2796	. . . . . . . . . . . . 13 ⊢ (¬ 𝐼 ∈ V → (𝑊 × 𝑊) = ∅)
42		ss0b 4424	. . . . . . . . . . . . 13 ⊢ ((𝑊 × 𝑊) ⊆ ∅ ↔ (𝑊 × 𝑊) = ∅)
43	41, 42	sylibr 234	. . . . . . . . . . . 12 ⊢ (¬ 𝐼 ∈ V → (𝑊 × 𝑊) ⊆ ∅)
44	34, 43	sylan9ssr 4023	. . . . . . . . . . 11 ⊢ ((¬ 𝐼 ∈ V ∧ 𝑟 Er 𝑊) → 𝑟 ⊆ ∅)
45	44	ex 412	. . . . . . . . . 10 ⊢ (¬ 𝐼 ∈ V → (𝑟 Er 𝑊 → 𝑟 ⊆ ∅))
46	45	adantrd 491	. . . . . . . . 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 1926	. . . . . . . 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 4034	. . . . . . . . 9 ⊢ (𝑤 = 𝑟 → (𝑤 ⊆ ∅ ↔ 𝑟 ⊆ ∅))
49	48	ralab2 3719	. . . . . . . 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 234	. . . . . . 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 4963	. . . . . . 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 234	. . . . . 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	sstrid 4020	. . . . 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 4425	. . . . 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 2783	. . 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 182	. 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 2768	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 206   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717   ≠ wne 2946  ∀wral 3067  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  ∅c0 4352  ⟨cop 4654  ⟨cotp 4656  ∪ cuni 4931  ∩ cint 4970   class class class wbr 5166   I cid 5592   × cxp 5698  Oncon0 6395  ‘cfv 6573  (class class class)co 7448  1_oc1o 8515  2_oc2o 8516   Er wer 8760  0cc0 11184  ...cfz 13567  ♯chash 14379  Word cword 14562   splice csplice 14797  ⟨“cs2 14890   ~_FG cefg 19748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-concat 14619  df-s1 14644  df-substr 14689  df-pfx 14719  df-splice 14798  df-s2 14897  df-efg 19751

This theorem is referenced by:  efger  19760  efgi  19761  efgval2  19766  frgpuplem  19814