Theorem efgrelexlemb 19971

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 × 2o))
efgval.r	⊢ ∼ = ( ~FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (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 × 2o))
2		efgval.r	. . 3 ⊢ ∼ = ( ~FG ‘𝐼)
3		efgval2.m	. . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
4		efgval2.t	. . 3 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
5	1, 2, 3, 4	efgval2 19945	. 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 2767	. . . . . . . . . 10 ⊢ ((𝑎‘0) = (𝑏‘0) ↔ (𝑏‘0) = (𝑎‘0))
10	9	2rexbii 3138	. . . . . . . . 9 ⊢ (∃𝑎 ∈ (◡𝑆 “ {𝑓})∃𝑏 ∈ (◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ (◡𝑆 “ {𝑓})∃𝑏 ∈ (◡𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0))
11		rexcom 3291	. . . . . . . . 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 19970	. . . . . . . 8 ⊢ (𝑓𝐿𝑔 ↔ ∃𝑎 ∈ (◡𝑆 “ {𝑓})∃𝑏 ∈ (◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0))
16	1, 2, 3, 4, 13, 14, 6	efgrelexlema 19970	. . . . . . . 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 19970	. . . . . . . . 9 ⊢ (𝑔𝐿ℎ ↔ ∃𝑟 ∈ (◡𝑆 “ {𝑔})∃𝑠 ∈ (◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0))
20		reeanv 3234	. . . . . . . . . 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 19960	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑆:dom 𝑆–onto→𝑊
22		fofn 6801	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆:dom 𝑆–onto→𝑊 → 𝑆 Fn dom 𝑆)
23	21, 22	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑆 Fn dom 𝑆
24		fniniseg 7062	. . . . . . . . . . . . . . . . . . 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 7062	. . . . . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆‘𝑟) = 𝑔 ∧ (𝑆‘𝑏) = 𝑔) → (𝑆‘𝑟) = (𝑆‘𝑏))
29	1, 2, 3, 4, 13, 14	efgred 19969	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = (𝑆‘𝑏)) → (𝑟‘0) = (𝑏‘0))
30	29	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∈ (◡𝑆 “ {𝑔}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑔})) → (∃𝑠 ∈ (◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (◡𝑆 “ {ℎ})(𝑏‘0) = (𝑠‘0)))
38		eqeq1 2764	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎‘0) = (𝑏‘0) → ((𝑎‘0) = (𝑠‘0) ↔ (𝑏‘0) = (𝑠‘0)))
39	38	rexbidv 3186	. . . . . . . . . . . . . . . 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 3163	. . . . . . . . . . . . 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 3156	. . . . . . . . . . 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 3100	. . . . . . . . . 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 19970	. . . . . . . 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 2760	. . . . . . . . . . . 12 ⊢ (𝑎‘0) = (𝑎‘0)
52		fveq1 6887	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑎 → (𝑏‘0) = (𝑎‘0))
53	52	rspceeqv 3598	. . . . . . . . . . . 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 7062	. . . . . . . . . . 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 3105	. . . . . . . 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 19970	. . . . . . . 8 ⊢ (𝑓𝐿𝑓 ↔ ∃𝑎 ∈ (◡𝑆 “ {𝑓})∃𝑏 ∈ (◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
61		forn 6802	. . . . . . . . . . 11 ⊢ (𝑆:dom 𝑆–onto→𝑊 → ran 𝑆 = 𝑊)
62	21, 61	ax-mp 5	. . . . . . . . . 10 ⊢ ran 𝑆 = 𝑊
63	62	eleq2i 2852	. . . . . . . . 9 ⊢ (𝑓 ∈ ran 𝑆 ↔ 𝑓 ∈ 𝑊)
64		fvelrnb 6948	. . . . . . . . . 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 8749	. . . . 5 ⊢ (⊤ → 𝐿 Er 𝑊)
70	69	mptru 1576	. . . 4 ⊢ 𝐿 Er 𝑊
71		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑎 ∈ 𝑊)
72		foelrn 7110	. . . . . . . . . . 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 2766	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑆‘𝑟) = 𝑎)
77		fniniseg 7062	. . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . 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 19958	. . . . . . . . . . . . 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 19951	. . . . . . . . . . . . . . . 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 4050	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ Word 𝑊)
90	1, 2, 3, 4	efgtf 19943	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ 𝑊 → ((𝑇‘𝑎) = (𝑓 ∈ (0...(♯‘𝑎)), 𝑔 ∈ (𝐼 × 2o) ↦ (𝑎 splice ⟨𝑓, 𝑓, ⟨“𝑔(𝑀‘𝑔)”⟩⟩)) ∧ (𝑇‘𝑎):((0...(♯‘𝑎)) × (𝐼 × 2o))⟶𝑊))
91	90	simprd 501	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝑊 → (𝑇‘𝑎):((0...(♯‘𝑎)) × (𝐼 × 2o))⟶𝑊)
92	91	frnd 6721	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝑊 → ran (𝑇‘𝑎) ⊆ 𝑊)
93	92	sselda 3909	. . . . . . . . . . . . . 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 19954	. . . . . . . . . . . . 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 7062	. . . . . . . . . . . . 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 14755	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → ⟨“𝑏”⟩ ∈ Word 𝑊)
101		eldifsn 4744	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ (Word 𝑊 ∖ {∅}) ↔ (𝑟 ∈ Word 𝑊 ∧ 𝑟 ≠ ∅))
102		lennncl 14684	. . . . . . . . . . . . . . . 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 13839	. . . . . . . . . . . . . 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 14727	. . . . . . . . . . . . 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 2766	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
110		fveq1 6887	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑟 ++ ⟨“𝑏”⟩) → (𝑠‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
111	110	rspceeqv 3598	. . . . . . . . . . 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 3180	. . . . . . . . 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 19970	. . . . . . . . 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 3454	. . . . . . . . 9 ⊢ 𝑏 ∈ V
117		vex 3454	. . . . . . . . 9 ⊢ 𝑎 ∈ V
118	116, 117	elec 8769	. . . . . . . 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 3915	. . . . 5 ⊢ (𝑎 ∈ 𝑊 → ran (𝑇‘𝑎) ⊆ [𝑎]𝐿)
122	121	rgen 3078	. . . 4 ⊢ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝐿
123	1	fvexi 6902	. . . . . 6 ⊢ 𝑊 ∈ V
124		erex 8747	. . . . . 6 ⊢ (𝐿 Er 𝑊 → (𝑊 ∈ V → 𝐿 ∈ V))
125	70, 123, 124	mp2 9	. . . . 5 ⊢ 𝐿 ∈ V
126		ereq1 8730	. . . . . 6 ⊢ (𝑟 = 𝐿 → (𝑟 Er 𝑊 ↔ 𝐿 Er 𝑊))
127		eceq2 8764	. . . . . . . 8 ⊢ (𝑟 = 𝐿 → [𝑎]𝑟 = [𝑎]𝐿)
128	127	sseq2d 3941	. . . . . . 7 ⊢ (𝑟 = 𝐿 → (ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 ↔ ran (𝑇‘𝑎) ⊆ [𝑎]𝐿))
129	128	ralbidv 3185	. . . . . 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 3632	. . . 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 4922	. . 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 3955	1 ⊢ ∼ ⊆ 𝐿

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ⊤wtru 1570   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3896   ∖ cdif 4044  ∅c0 4277  {csn 4584  ⟨cop 4834  ⟨cotp 4857  ∩ cint 4906  ∪ ciun 4950   class class class wbr 5102  {copab 5166   ↦ cmpt 5185   I cid 5544   × cxp 5648  ^◡ccnv 5649  dom cdm 5650  ran crn 5651   “ cima 5653  Rel wrel 5655   Fn wfn 6536  ⟶wf 6537  –onto→wfo 6539  ‘cfv 6541  (class class class)co 7422   ∈ cmpo 7424  1_oc1o 8474  2_oc2o 8475   Er wer 8719  [cec 8720  0cc0 11205  1c1 11206   − cmin 11546  ℕcn 12340  ...cfz 13644  ..^cfzo 13793  ♯chash 14479  Word cword 14663   ++ cconcat 14720  ⟨“cs1 14747   splice csplice 14904  ⟨“cs2 14998   ~_FG cefg 19927

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-cnex 11261  ax-resscn 11262  ax-1cn 11263  ax-icn 11264  ax-addcl 11265  ax-addrcl 11266  ax-mulcl 11267  ax-mulrcl 11268  ax-mulcom 11269  ax-addass 11270  ax-mulass 11271  ax-distr 11272  ax-i2m1 11273  ax-1ne0 11274  ax-1rid 11275  ax-rnegex 11276  ax-rrecex 11277  ax-cnre 11278  ax-pre-lttri 11279  ax-pre-lttrn 11280  ax-pre-ltadd 11281  ax-pre-mulgt0 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-om 7880  df-1st 8003  df-2nd 8004  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-1o 8481  df-2o 8482  df-er 8722  df-ec 8724  df-map 8854  df-en 8979  df-dom 8980  df-sdom 8981  df-fin 8982  df-card 10025  df-pnf 11350  df-mnf 11351  df-xr 11352  df-ltxr 11353  df-le 11354  df-sub 11548  df-neg 11549  df-nn 12341  df-2 12410  df-n0 12612  df-xnn0 12685  df-z 12699  df-uz 12971  df-rp 13126  df-fz 13645  df-fzo 13794  df-hash 14480  df-word 14664  df-concat 14721  df-s1 14748  df-substr 14795  df-pfx 14827  df-splice 14905  df-s2 15005  df-efg 19930

This theorem is used by:  efgrelex  19972