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Theorem efgrelexlemb 19957

Description: If two words 𝐴, 𝐵 are related under the free group equivalence, then there exist two extension sequences 𝑎, 𝑏 such that 𝑎 ends at 𝐴, 𝑏 ends at 𝐵, and 𝑎 and 𝐵 have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.)

**Hypotheses**
Ref	Expression
efgval.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o))
efgval.r	⊢ ∼ = ( ~_FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
efgred.d	⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
efgred.s	⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
efgrelexlem.1	⊢ 𝐿 = {⟨𝑖, 𝑗⟩ ∣ ∃𝑐 ∈ (^◡𝑆 “ {𝑖})∃𝑑 ∈ (^◡𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0)}

**Assertion**
Ref	Expression
efgrelexlemb	⊢ ∼ ⊆ 𝐿

Distinct variable groups: 𝑐,𝑑,𝑖,𝑗 𝑦,𝑧 𝑛,𝑐,𝑡,𝑣,𝑤,𝑦,𝑧,𝑚,𝑥 𝑀,𝑐 𝑖,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑀,𝑗 𝑘,𝑐,𝑇,𝑖,𝑗,𝑚,𝑡,𝑥 𝑊,𝑐 𝑘,𝑑,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧,𝑊,𝑖,𝑗 ∼ ,𝑐,𝑑,𝑖,𝑗,𝑚,𝑡,𝑥,𝑦,𝑧 𝑆,𝑐,𝑑,𝑖,𝑗 𝐼,𝑐,𝑖,𝑗,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧 𝐷,𝑐,𝑑,𝑖,𝑗,𝑚,𝑡 Allowed substitution hints: 𝐷(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑘, 𝑛) ∼ (𝑤, 𝑣, 𝑘, 𝑛) 𝑆(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑡, 𝑘, 𝑚, 𝑛) 𝑇(𝑦, 𝑧, 𝑤, 𝑣, 𝑛, 𝑑) 𝐼(𝑘, 𝑑) 𝐿(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑡, 𝑖, 𝑗, 𝑘, 𝑚, 𝑛, 𝑐, 𝑑) 𝑀(𝑦, 𝑧, 𝑘, 𝑑)

Proof of Theorem efgrelexlemb Dummy variables 𝑎 𝑏 𝑓 𝑔 ℎ 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . 3 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o))
2		efgval.r	. . 3 ⊢ ∼ = ( ~_FG ‘𝐼)
3		efgval2.m	. . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)
4		efgval2.t	. . 3 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
5	1, 2, 3, 4	efgval2 19931	. 2 ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟)}
6		efgrelexlem.1	. . . . . . . 8 ⊢ 𝐿 = {⟨𝑖, 𝑗⟩ ∣ ∃𝑐 ∈ (^◡𝑆 “ {𝑖})∃𝑑 ∈ (^◡𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0)}
7	6	relopabiv 5798	. . . . . . 7 ⊢ Rel 𝐿
8	7	a1i 11	. . . . . 6 ⊢ (⊤ → Rel 𝐿)
9		eqcom 2768	. . . . . . . . . 10 ⊢ ((𝑎‘0) = (𝑏‘0) ↔ (𝑏‘0) = (𝑎‘0))
10	9	2rexbii 3139	. . . . . . . . 9 ⊢ (∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0))
11		rexcom 3292	. . . . . . . . 9 ⊢ (∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0) ↔ ∃𝑏 ∈ (^◡𝑆 “ {𝑔})∃𝑎 ∈ (^◡𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
12	10, 11	bitri 278	. . . . . . . 8 ⊢ (∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑏 ∈ (^◡𝑆 “ {𝑔})∃𝑎 ∈ (^◡𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
13		efgred.d	. . . . . . . . 9 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
14		efgred.s	. . . . . . . . 9 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
15	1, 2, 3, 4, 13, 14, 6	efgrelexlema 19956	. . . . . . . 8 ⊢ (𝑓𝐿𝑔 ↔ ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0))
16	1, 2, 3, 4, 13, 14, 6	efgrelexlema 19956	. . . . . . . 8 ⊢ (𝑔𝐿𝑓 ↔ ∃𝑏 ∈ (^◡𝑆 “ {𝑔})∃𝑎 ∈ (^◡𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
17	12, 15, 16	3bitr4i 306	. . . . . . 7 ⊢ (𝑓𝐿𝑔 ↔ 𝑔𝐿𝑓)
18	17	bilani 510	. . . . . 6 ⊢ ((⊤ ∧ 𝑓𝐿𝑔) → 𝑔𝐿𝑓)
19	1, 2, 3, 4, 13, 14, 6	efgrelexlema 19956	. . . . . . . . 9 ⊢ (𝑔𝐿ℎ ↔ ∃𝑟 ∈ (^◡𝑆 “ {𝑔})∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0))
20		reeanv 3235	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑟 ∈ (^◡𝑆 “ {𝑔})(∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0)) ↔ (∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑟 ∈ (^◡𝑆 “ {𝑔})∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0)))
21	1, 2, 3, 4, 13, 14	efgsfo 19946	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑆:dom 𝑆–onto→𝑊
22		fofn 6796	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆:dom 𝑆–onto→𝑊 → 𝑆 Fn dom 𝑆)
23	21, 22	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑆 Fn dom 𝑆
24		fniniseg 7057	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 Fn dom 𝑆 → (𝑟 ∈ (^◡𝑆 “ {𝑔}) ↔ (𝑟 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = 𝑔)))
25	23, 24	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ (^◡𝑆 “ {𝑔}) ↔ (𝑟 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = 𝑔))
26		fniniseg 7057	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 Fn dom 𝑆 → (𝑏 ∈ (^◡𝑆 “ {𝑔}) ↔ (𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑏) = 𝑔)))
27	23, 26	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ (^◡𝑆 “ {𝑔}) ↔ (𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑏) = 𝑔))
28		eqtr3 2783	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆‘𝑟) = 𝑔 ∧ (𝑆‘𝑏) = 𝑔) → (𝑆‘𝑟) = (𝑆‘𝑏))
29	1, 2, 3, 4, 13, 14	efgred 19955	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = (𝑆‘𝑏)) → (𝑟‘0) = (𝑏‘0))
30	29	eqcomd 2767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = (𝑆‘𝑏)) → (𝑏‘0) = (𝑟‘0))
31	30	3expa 1136	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆) ∧ (𝑆‘𝑟) = (𝑆‘𝑏)) → (𝑏‘0) = (𝑟‘0))
32	28, 31	sylan2 605	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆) ∧ ((𝑆‘𝑟) = 𝑔 ∧ (𝑆‘𝑏) = 𝑔)) → (𝑏‘0) = (𝑟‘0))
33	32	an4s 673	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑟 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = 𝑔) ∧ (𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑏) = 𝑔)) → (𝑏‘0) = (𝑟‘0))
34	25, 27, 33	syl2anb 610	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑟 ∈ (^◡𝑆 “ {𝑔}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑔})) → (𝑏‘0) = (𝑟‘0))
35		eqeq2 2773	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑟‘0) = (𝑠‘0) → ((𝑏‘0) = (𝑟‘0) ↔ (𝑏‘0) = (𝑠‘0)))
36	34, 35	syl5ibcom 248	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ (^◡𝑆 “ {𝑔}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑔})) → ((𝑟‘0) = (𝑠‘0) → (𝑏‘0) = (𝑠‘0)))
37	36	reximdv 3178	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∈ (^◡𝑆 “ {𝑔}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑔})) → (∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑏‘0) = (𝑠‘0)))
38		eqeq1 2765	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎‘0) = (𝑏‘0) → ((𝑎‘0) = (𝑠‘0) ↔ (𝑏‘0) = (𝑠‘0)))
39	38	rexbidv 3187	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0) ↔ ∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑏‘0) = (𝑠‘0)))
40	39	imbi2d 343	. . . . . . . . . . . . . . 15 ⊢ ((𝑎‘0) = (𝑏‘0) → ((∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0)) ↔ (∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑏‘0) = (𝑠‘0))))
41	37, 40	syl5ibrcom 250	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∈ (^◡𝑆 “ {𝑔}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑔})) → ((𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0))))
42	41	rexlimdva 3164	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ (^◡𝑆 “ {𝑔}) → (∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0))))
43	42	impd 416	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ (^◡𝑆 “ {𝑔}) → ((∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0)) → ∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0)))
44	43	rexlimiv 3157	. . . . . . . . . . 11 ⊢ (∃𝑟 ∈ (^◡𝑆 “ {𝑔})(∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0)) → ∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0))
45	44	reximi 3101	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑟 ∈ (^◡𝑆 “ {𝑔})(∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0)) → ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0))
46	20, 45	sylbir 238	. . . . . . . . 9 ⊢ ((∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ∧ ∃𝑟 ∈ (^◡𝑆 “ {𝑔})∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0)) → ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0))
47	15, 19, 46	syl2anb 610	. . . . . . . 8 ⊢ ((𝑓𝐿𝑔 ∧ 𝑔𝐿ℎ) → ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0))
48	1, 2, 3, 4, 13, 14, 6	efgrelexlema 19956	. . . . . . . 8 ⊢ (𝑓𝐿ℎ ↔ ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0))
49	47, 48	sylibr 237	. . . . . . 7 ⊢ ((𝑓𝐿𝑔 ∧ 𝑔𝐿ℎ) → 𝑓𝐿ℎ)
50	49	adantl 487	. . . . . 6 ⊢ ((⊤ ∧ (𝑓𝐿𝑔 ∧ 𝑔𝐿ℎ)) → 𝑓𝐿ℎ)
51		eqid 2761	. . . . . . . . . . . 12 ⊢ (𝑎‘0) = (𝑎‘0)
52		fveq1 6882	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑎 → (𝑏‘0) = (𝑎‘0))
53	52	rspceeqv 3599	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ (^◡𝑆 “ {𝑓}) ∧ (𝑎‘0) = (𝑎‘0)) → ∃𝑏 ∈ (^◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
54	51, 53	mpan2 704	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (^◡𝑆 “ {𝑓}) → ∃𝑏 ∈ (^◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
55	54	pm4.71i 569	. . . . . . . . . 10 ⊢ (𝑎 ∈ (^◡𝑆 “ {𝑓}) ↔ (𝑎 ∈ (^◡𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (^◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)))
56		fniniseg 7057	. . . . . . . . . . 11 ⊢ (𝑆 Fn dom 𝑆 → (𝑎 ∈ (^◡𝑆 “ {𝑓}) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆‘𝑎) = 𝑓)))
57	23, 56	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑎 ∈ (^◡𝑆 “ {𝑓}) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆‘𝑎) = 𝑓))
58	55, 57	bitr3i 280	. . . . . . . . 9 ⊢ ((𝑎 ∈ (^◡𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (^◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆‘𝑎) = 𝑓))
59	58	rexbii2 3106	. . . . . . . 8 ⊢ (∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ dom 𝑆(𝑆‘𝑎) = 𝑓)
60	1, 2, 3, 4, 13, 14, 6	efgrelexlema 19956	. . . . . . . 8 ⊢ (𝑓𝐿𝑓 ↔ ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
61		forn 6797	. . . . . . . . . . 11 ⊢ (𝑆:dom 𝑆–onto→𝑊 → ran 𝑆 = 𝑊)
62	21, 61	ax-mp 5	. . . . . . . . . 10 ⊢ ran 𝑆 = 𝑊
63	62	eleq2i 2853	. . . . . . . . 9 ⊢ (𝑓 ∈ ran 𝑆 ↔ 𝑓 ∈ 𝑊)
64		fvelrnb 6943	. . . . . . . . . 10 ⊢ (𝑆 Fn dom 𝑆 → (𝑓 ∈ ran 𝑆 ↔ ∃𝑎 ∈ dom 𝑆(𝑆‘𝑎) = 𝑓))
65	23, 64	ax-mp 5	. . . . . . . . 9 ⊢ (𝑓 ∈ ran 𝑆 ↔ ∃𝑎 ∈ dom 𝑆(𝑆‘𝑎) = 𝑓)
66	63, 65	bitr3i 280	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑊 ↔ ∃𝑎 ∈ dom 𝑆(𝑆‘𝑎) = 𝑓)
67	59, 60, 66	3bitr4ri 307	. . . . . . 7 ⊢ (𝑓 ∈ 𝑊 ↔ 𝑓𝐿𝑓)
68	67	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑓 ∈ 𝑊 ↔ 𝑓𝐿𝑓))
69	8, 18, 50, 68	iserd 8737	. . . . 5 ⊢ (⊤ → 𝐿 Er 𝑊)
70	69	mptru 1577	. . . 4 ⊢ 𝐿 Er 𝑊
71		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑎 ∈ 𝑊)
72		foelrn 7105	. . . . . . . . . . 11 ⊢ ((𝑆:dom 𝑆–onto→𝑊 ∧ 𝑎 ∈ 𝑊) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆‘𝑟))
73	21, 71, 72	sylancr 599	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆‘𝑟))
74		simprl 783	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ dom 𝑆)
75		simprr 785	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑎 = (𝑆‘𝑟))
76	75	eqcomd 2767	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑆‘𝑟) = 𝑎)
77		fniniseg 7057	. . . . . . . . . . . 12 ⊢ (𝑆 Fn dom 𝑆 → (𝑟 ∈ (^◡𝑆 “ {𝑎}) ↔ (𝑟 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = 𝑎)))
78	23, 77	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑟 ∈ (^◡𝑆 “ {𝑎}) ↔ (𝑟 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = 𝑎))
79	74, 76, 78	sylanbrc 595	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ (^◡𝑆 “ {𝑎}))
80		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑏 ∈ ran (𝑇‘𝑎))
81	75	fveq2d 6887	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑇‘𝑎) = (𝑇‘(𝑆‘𝑟)))
82	81	rneqd 5920	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → ran (𝑇‘𝑎) = ran (𝑇‘(𝑆‘𝑟)))
83	80, 82	eleqtrd 2863	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑏 ∈ ran (𝑇‘(𝑆‘𝑟)))
84	1, 2, 3, 4, 13, 14	efgsp1 19944	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ ran (𝑇‘(𝑆‘𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆)
85	74, 83, 84	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆)
86	1, 2, 3, 4, 13, 14	efgsdm 19937	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ dom 𝑆 ↔ (𝑟 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑟‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑟))(𝑟‘𝑖) ∈ ran (𝑇‘(𝑟‘(𝑖 − 1)))))
87	86	simp1bi 1163	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ dom 𝑆 → 𝑟 ∈ (Word 𝑊 ∖ {∅}))
88	87	ad2antrl 741	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ (Word 𝑊 ∖ {∅}))
89	88	eldifad 3911	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ Word 𝑊)
90	1, 2, 3, 4	efgtf 19929	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ 𝑊 → ((𝑇‘𝑎) = (𝑓 ∈ (0...(♯‘𝑎)), 𝑔 ∈ (𝐼 × 2_o) ↦ (𝑎 splice ⟨𝑓, 𝑓, ⟨“𝑔(𝑀‘𝑔)”⟩⟩)) ∧ (𝑇‘𝑎):((0...(♯‘𝑎)) × (𝐼 × 2_o))⟶𝑊))
91	90	simprd 501	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝑊 → (𝑇‘𝑎):((0...(♯‘𝑎)) × (𝐼 × 2_o))⟶𝑊)
92	91	frnd 6716	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝑊 → ran (𝑇‘𝑎) ⊆ 𝑊)
93	92	sselda 3931	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑏 ∈ 𝑊)
94	93	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑏 ∈ 𝑊)
95	1, 2, 3, 4, 13, 14	efgsval2 19940	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ Word 𝑊 ∧ 𝑏 ∈ 𝑊 ∧ (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
96	89, 94, 85, 95	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
97		fniniseg 7057	. . . . . . . . . . . . 13 ⊢ (𝑆 Fn dom 𝑆 → ((𝑟 ++ ⟨“𝑏”⟩) ∈ (^◡𝑆 “ {𝑏}) ↔ ((𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆 ∧ (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)))
98	23, 97	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑟 ++ ⟨“𝑏”⟩) ∈ (^◡𝑆 “ {𝑏}) ↔ ((𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆 ∧ (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏))
99	85, 96, 98	sylanbrc 595	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ (^◡𝑆 “ {𝑏}))
100	94	s1cld 14743	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → ⟨“𝑏”⟩ ∈ Word 𝑊)
101		eldifsn 4748	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ (Word 𝑊 ∖ {∅}) ↔ (𝑟 ∈ Word 𝑊 ∧ 𝑟 ≠ ∅))
102		lennncl 14672	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ Word 𝑊 ∧ 𝑟 ≠ ∅) → (♯‘𝑟) ∈ ℕ)
103	101, 102	sylbi 220	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ (Word 𝑊 ∖ {∅}) → (♯‘𝑟) ∈ ℕ)
104	88, 103	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (♯‘𝑟) ∈ ℕ)
105		lbfzo0 13827	. . . . . . . . . . . . . 14 ⊢ (0 ∈ (0..^(♯‘𝑟)) ↔ (♯‘𝑟) ∈ ℕ)
106	104, 105	sylibr 237	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 0 ∈ (0..^(♯‘𝑟)))
107		ccatval1 14715	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ Word 𝑊 ∧ ⟨“𝑏”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(♯‘𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
108	89, 100, 106, 107	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
109	108	eqcomd 2767	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
110		fveq1 6882	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑟 ++ ⟨“𝑏”⟩) → (𝑠‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
111	110	rspceeqv 3599	. . . . . . . . . . 11 ⊢ (((𝑟 ++ ⟨“𝑏”⟩) ∈ (^◡𝑆 “ {𝑏}) ∧ (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0)) → ∃𝑠 ∈ (^◡𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
112	99, 109, 111	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → ∃𝑠 ∈ (^◡𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
113	73, 79, 112	reximssdv 3181	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → ∃𝑟 ∈ (^◡𝑆 “ {𝑎})∃𝑠 ∈ (^◡𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
114	1, 2, 3, 4, 13, 14, 6	efgrelexlema 19956	. . . . . . . . 9 ⊢ (𝑎𝐿𝑏 ↔ ∃𝑟 ∈ (^◡𝑆 “ {𝑎})∃𝑠 ∈ (^◡𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
115	113, 114	sylibr 237	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑎𝐿𝑏)
116		vex 3455	. . . . . . . . 9 ⊢ 𝑏 ∈ V
117		vex 3455	. . . . . . . . 9 ⊢ 𝑎 ∈ V
118	116, 117	elec 8757	. . . . . . . 8 ⊢ (𝑏 ∈ [𝑎]𝐿 ↔ 𝑎𝐿𝑏)
119	115, 118	sylibr 237	. . . . . . 7 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑏 ∈ [𝑎]𝐿)
120	119	ex 418	. . . . . 6 ⊢ (𝑎 ∈ 𝑊 → (𝑏 ∈ ran (𝑇‘𝑎) → 𝑏 ∈ [𝑎]𝐿))
121	120	ssrdv 3937	. . . . 5 ⊢ (𝑎 ∈ 𝑊 → ran (𝑇‘𝑎) ⊆ [𝑎]𝐿)
122	121	rgen 3079	. . . 4 ⊢ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝐿
123	1	fvexi 6897	. . . . . 6 ⊢ 𝑊 ∈ V
124		erex 8735	. . . . . 6 ⊢ (𝐿 Er 𝑊 → (𝑊 ∈ V → 𝐿 ∈ V))
125	70, 123, 124	mp2 9	. . . . 5 ⊢ 𝐿 ∈ V
126		ereq1 8718	. . . . . 6 ⊢ (𝑟 = 𝐿 → (𝑟 Er 𝑊 ↔ 𝐿 Er 𝑊))
127		eceq2 8752	. . . . . . . 8 ⊢ (𝑟 = 𝐿 → [𝑎]𝑟 = [𝑎]𝐿)
128	127	sseq2d 3963	. . . . . . 7 ⊢ (𝑟 = 𝐿 → (ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 ↔ ran (𝑇‘𝑎) ⊆ [𝑎]𝐿))
129	128	ralbidv 3186	. . . . . 6 ⊢ (𝑟 = 𝐿 → (∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 ↔ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝐿))
130	126, 129	anbi12d 644	. . . . 5 ⊢ (𝑟 = 𝐿 → ((𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟) ↔ (𝐿 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝐿)))
131	125, 130	elab 3633	. . . 4 ⊢ (𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟)} ↔ (𝐿 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝐿))
132	70, 122, 131	mpbir2an 724	. . 3 ⊢ 𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟)}
133		intss1 4923	. . 3 ⊢ (𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟)} → ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟)} ⊆ 𝐿)
134	132, 133	ax-mp 5	. 2 ⊢ ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟)} ⊆ 𝐿
135	5, 134	eqsstri 3977	1 ⊢ ∼ ⊆ 𝐿

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ⊤wtru 1571 ∈ wcel 2145 {cab 2739 ≠ wne 2956 ∀wral 3077 ∃wrex 3087 {crab 3413 Vcvv 3451 ∖ cdif 3896 ⊆ wss 3899 ∅c0 4279 {csn 4584 ⟨cop 4590 ⟨cotp 4592 ∩ cint 4907 ∪ ciun 4951 class class class wbr 5103 {copab 5167 ↦ cmpt 5186 I cid 5545 × cxp 5649 ^◡ccnv 5650 dom cdm 5651 ran crn 5652 “ cima 5654 Rel wrel 5656 Fn wfn 6532 ⟶wf 6533 –onto→wfo 6535 ‘cfv 6537 (class class class)co 7418 ∈ cmpo 7420 1_oc1o 8462 2_oc2o 8463 Er wer 8707 [cec 8708 0cc0 11193 1c1 11194 − cmin 11534 ℕcn 12328 ...cfz 13632 ..^cfzo 13781 ♯chash 14467 Word cword 14651 ++ cconcat 14708 ⟨“cs1 14735 splice csplice 14891 ⟨“cs2 14985 ~_FG cefg 19913

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-er 8710 df-ec 8712 df-map 8842 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-nn 12329 df-2 12398 df-n0 12600 df-xnn0 12673 df-z 12687 df-uz 12959 df-rp 13114 df-fz 13633 df-fzo 13782 df-hash 14468 df-word 14652 df-concat 14709 df-s1 14736 df-substr 14782 df-pfx 14814 df-splice 14892 df-s2 14992 df-efg 19916

This theorem is used by: efgrelex 19958

W3C validator