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Theorem efgrelexlemb 19781

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 19755	. 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 5789	. . . . . . 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 3137	. . . . . . . . 9 ⊢ (∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0))
11		rexcom 3290	. . . . . . . . 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 277	. . . . . . . 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 19780	. . . . . . . 8 ⊢ (𝑓𝐿𝑔 ↔ ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0))
16	1, 2, 3, 4, 13, 14, 6	efgrelexlema 19780	. . . . . . . 8 ⊢ (𝑔𝐿𝑓 ↔ ∃𝑏 ∈ (^◡𝑆 “ {𝑔})∃𝑎 ∈ (^◡𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
17	12, 15, 16	3bitr4i 305	. . . . . . 7 ⊢ (𝑓𝐿𝑔 ↔ 𝑔𝐿𝑓)
18	17	bilani 508	. . . . . 6 ⊢ ((⊤ ∧ 𝑓𝐿𝑔) → 𝑔𝐿𝑓)
19	1, 2, 3, 4, 13, 14, 6	efgrelexlema 19780	. . . . . . . . 9 ⊢ (𝑔𝐿ℎ ↔ ∃𝑟 ∈ (^◡𝑆 “ {𝑔})∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0))
20		reeanv 3233	. . . . . . . . . 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 19770	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑆:dom 𝑆–onto→𝑊
22		fofn 6775	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆:dom 𝑆–onto→𝑊 → 𝑆 Fn dom 𝑆)
23	21, 22	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑆 Fn dom 𝑆
24		fniniseg 7036	. . . . . . . . . . . . . . . . . . 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 7036	. . . . . . . . . . . . . . . . . . 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 19779	. . . . . . . . . . . . . . . . . . . . . 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 1130	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆) ∧ (𝑆‘𝑟) = (𝑆‘𝑏)) → (𝑏‘0) = (𝑟‘0))
32	28, 31	sylan2 602	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆) ∧ ((𝑆‘𝑟) = 𝑔 ∧ (𝑆‘𝑏) = 𝑔)) → (𝑏‘0) = (𝑟‘0))
33	32	an4s 670	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑟 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = 𝑔) ∧ (𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑏) = 𝑔)) → (𝑏‘0) = (𝑟‘0))
34	25, 27, 33	syl2anb 607	. . . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ (^◡𝑆 “ {𝑔}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑔})) → ((𝑟‘0) = (𝑠‘0) → (𝑏‘0) = (𝑠‘0)))
37	36	reximdv 3176	. . . . . . . . . . . . . . 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 3185	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0) ↔ ∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑏‘0) = (𝑠‘0)))
40	39	imbi2d 342	. . . . . . . . . . . . . . 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 249	. . . . . . . . . . . . . 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 3162	. . . . . . . . . . . . 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 414	. . . . . . . . . . . 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 3155	. . . . . . . . . . 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 3099	. . . . . . . . . 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 237	. . . . . . . . 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 607	. . . . . . . 8 ⊢ ((𝑓𝐿𝑔 ∧ 𝑔𝐿ℎ) → ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0))
48	1, 2, 3, 4, 13, 14, 6	efgrelexlema 19780	. . . . . . . 8 ⊢ (𝑓𝐿ℎ ↔ ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0))
49	47, 48	sylibr 236	. . . . . . 7 ⊢ ((𝑓𝐿𝑔 ∧ 𝑔𝐿ℎ) → 𝑓𝐿ℎ)
50	49	adantl 485	. . . . . 6 ⊢ ((⊤ ∧ (𝑓𝐿𝑔 ∧ 𝑔𝐿ℎ)) → 𝑓𝐿ℎ)
51		eqid 2761	. . . . . . . . . . . 12 ⊢ (𝑎‘0) = (𝑎‘0)
52		fveq1 6861	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑎 → (𝑏‘0) = (𝑎‘0))
53	52	rspceeqv 3603	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ (^◡𝑆 “ {𝑓}) ∧ (𝑎‘0) = (𝑎‘0)) → ∃𝑏 ∈ (^◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
54	51, 53	mpan2 701	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (^◡𝑆 “ {𝑓}) → ∃𝑏 ∈ (^◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
55	54	pm4.71i 567	. . . . . . . . . 10 ⊢ (𝑎 ∈ (^◡𝑆 “ {𝑓}) ↔ (𝑎 ∈ (^◡𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (^◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)))
56		fniniseg 7036	. . . . . . . . . . 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 279	. . . . . . . . 9 ⊢ ((𝑎 ∈ (^◡𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (^◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆‘𝑎) = 𝑓))
59	58	rexbii2 3104	. . . . . . . 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 19780	. . . . . . . 8 ⊢ (𝑓𝐿𝑓 ↔ ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
61		forn 6776	. . . . . . . . . . 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 6922	. . . . . . . . . 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 279	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑊 ↔ ∃𝑎 ∈ dom 𝑆(𝑆‘𝑎) = 𝑓)
67	59, 60, 66	3bitr4ri 306	. . . . . . 7 ⊢ (𝑓 ∈ 𝑊 ↔ 𝑓𝐿𝑓)
68	67	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑓 ∈ 𝑊 ↔ 𝑓𝐿𝑓))
69	8, 18, 50, 68	iserd 8699	. . . . 5 ⊢ (⊤ → 𝐿 Er 𝑊)
70	69	mptru 1566	. . . 4 ⊢ 𝐿 Er 𝑊
71		simpl 486	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑎 ∈ 𝑊)
72		foelrn 7083	. . . . . . . . . . 11 ⊢ ((𝑆:dom 𝑆–onto→𝑊 ∧ 𝑎 ∈ 𝑊) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆‘𝑟))
73	21, 71, 72	sylancr 596	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆‘𝑟))
74		simprl 780	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ dom 𝑆)
75		simprr 782	. . . . . . . . . . . 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 7036	. . . . . . . . . . . 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 592	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ (^◡𝑆 “ {𝑎}))
80		simplr 778	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑏 ∈ ran (𝑇‘𝑎))
81	75	fveq2d 6866	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑇‘𝑎) = (𝑇‘(𝑆‘𝑟)))
82	81	rneqd 5910	. . . . . . . . . . . . . 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 19768	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ ran (𝑇‘(𝑆‘𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆)
85	74, 83, 84	syl2anc 593	. . . . . . . . . . . 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 19761	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ dom 𝑆 ↔ (𝑟 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑟‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑟))(𝑟‘𝑖) ∈ ran (𝑇‘(𝑟‘(𝑖 − 1)))))
87	86	simp1bi 1157	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ dom 𝑆 → 𝑟 ∈ (Word 𝑊 ∖ {∅}))
88	87	ad2antrl 738	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ (Word 𝑊 ∖ {∅}))
89	88	eldifad 3914	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ Word 𝑊)
90	1, 2, 3, 4	efgtf 19753	. . . . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝑊 → (𝑇‘𝑎):((0...(♯‘𝑎)) × (𝐼 × 2_o))⟶𝑊)
92	91	frnd 6695	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝑊 → ran (𝑇‘𝑎) ⊆ 𝑊)
93	92	sselda 3934	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑏 ∈ 𝑊)
94	93	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑏 ∈ 𝑊)
95	1, 2, 3, 4, 13, 14	efgsval2 19764	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ Word 𝑊 ∧ 𝑏 ∈ 𝑊 ∧ (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
96	89, 94, 85, 95	syl3anc 1389	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
97		fniniseg 7036	. . . . . . . . . . . . 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 592	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ (^◡𝑆 “ {𝑏}))
100	94	s1cld 14611	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → ⟨“𝑏”⟩ ∈ Word 𝑊)
101		eldifsn 4743	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ (Word 𝑊 ∖ {∅}) ↔ (𝑟 ∈ Word 𝑊 ∧ 𝑟 ≠ ∅))
102		lennncl 14541	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ Word 𝑊 ∧ 𝑟 ≠ ∅) → (♯‘𝑟) ∈ ℕ)
103	101, 102	sylbi 219	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ (Word 𝑊 ∖ {∅}) → (♯‘𝑟) ∈ ℕ)
104	88, 103	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (♯‘𝑟) ∈ ℕ)
105		lbfzo0 13699	. . . . . . . . . . . . . 14 ⊢ (0 ∈ (0..^(♯‘𝑟)) ↔ (♯‘𝑟) ∈ ℕ)
106	104, 105	sylibr 236	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 0 ∈ (0..^(♯‘𝑟)))
107		ccatval1 14584	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ Word 𝑊 ∧ ⟨“𝑏”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(♯‘𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
108	89, 100, 106, 107	syl3anc 1389	. . . . . . . . . . . 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 6861	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑟 ++ ⟨“𝑏”⟩) → (𝑠‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
111	110	rspceeqv 3603	. . . . . . . . . . 11 ⊢ (((𝑟 ++ ⟨“𝑏”⟩) ∈ (^◡𝑆 “ {𝑏}) ∧ (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0)) → ∃𝑠 ∈ (^◡𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
112	99, 109, 111	syl2anc 593	. . . . . . . . . 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 3179	. . . . . . . . 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 19780	. . . . . . . . 9 ⊢ (𝑎𝐿𝑏 ↔ ∃𝑟 ∈ (^◡𝑆 “ {𝑎})∃𝑠 ∈ (^◡𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
115	113, 114	sylibr 236	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑎𝐿𝑏)
116		vex 3457	. . . . . . . . 9 ⊢ 𝑏 ∈ V
117		vex 3457	. . . . . . . . 9 ⊢ 𝑎 ∈ V
118	116, 117	elec 8719	. . . . . . . 8 ⊢ (𝑏 ∈ [𝑎]𝐿 ↔ 𝑎𝐿𝑏)
119	115, 118	sylibr 236	. . . . . . 7 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑏 ∈ [𝑎]𝐿)
120	119	ex 416	. . . . . 6 ⊢ (𝑎 ∈ 𝑊 → (𝑏 ∈ ran (𝑇‘𝑎) → 𝑏 ∈ [𝑎]𝐿))
121	120	ssrdv 3940	. . . . 5 ⊢ (𝑎 ∈ 𝑊 → ran (𝑇‘𝑎) ⊆ [𝑎]𝐿)
122	121	rgen 3077	. . . 4 ⊢ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝐿
123	1	fvexi 6876	. . . . . 6 ⊢ 𝑊 ∈ V
124		erex 8697	. . . . . 6 ⊢ (𝐿 Er 𝑊 → (𝑊 ∈ V → 𝐿 ∈ V))
125	70, 123, 124	mp2 9	. . . . 5 ⊢ 𝐿 ∈ V
126		ereq1 8680	. . . . . 6 ⊢ (𝑟 = 𝐿 → (𝑟 Er 𝑊 ↔ 𝐿 Er 𝑊))
127		eceq2 8714	. . . . . . . 8 ⊢ (𝑟 = 𝐿 → [𝑎]𝑟 = [𝑎]𝐿)
128	127	sseq2d 3966	. . . . . . 7 ⊢ (𝑟 = 𝐿 → (ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 ↔ ran (𝑇‘𝑎) ⊆ [𝑎]𝐿))
129	128	ralbidv 3184	. . . . . 6 ⊢ (𝑟 = 𝐿 → (∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 ↔ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝐿))
130	126, 129	anbi12d 641	. . . . 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 3637	. . . 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 721	. . 3 ⊢ 𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟)}
133		intss1 4918	. . 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 3980	1 ⊢ ∼ ⊆ 𝐿

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ⊤wtru 1560 ∈ wcel 2141 {cab 2739 ≠ wne 2956 ∀wral 3075 ∃wrex 3085 {crab 3413 Vcvv 3453 ∖ cdif 3899 ⊆ wss 3902 ∅c0 4283 {csn 4579 ⟨cop 4585 ⟨cotp 4587 ∩ cint 4902 ∪ ciun 4946 class class class wbr 5097 {copab 5159 ↦ cmpt 5178 I cid 5537 × cxp 5641 ^◡ccnv 5642 dom cdm 5643 ran crn 5644 “ cima 5646 Rel wrel 5648 Fn wfn 6511 ⟶wf 6512 –onto→wfo 6514 ‘cfv 6516 (class class class)co 7391 ∈ cmpo 7393 1_oc1o 8424 2_oc2o 8425 Er wer 8669 [cec 8670 0cc0 11067 1c1 11068 − cmin 11408 ℕcn 12204 ...cfz 13506 ..^cfzo 13653 ♯chash 14337 Word cword 14520 ++ cconcat 14577 ⟨“cs1 14603 splice csplice 14756 ⟨“cs2 14848 ~_FG cefg 19737

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-ot 4588 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-er 8672 df-ec 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-nn 12205 df-2 12274 df-n0 12476 df-xnn0 12549 df-z 12563 df-uz 12834 df-rp 12988 df-fz 13507 df-fzo 13654 df-hash 14338 df-word 14521 df-concat 14578 df-s1 14604 df-substr 14649 df-pfx 14679 df-splice 14757 df-s2 14855 df-efg 19740

This theorem is referenced by: efgrelex 19782

W3C validator