Theorem efgrelexlemb 19679

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 19653	. 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 5769	. . . . . . 7 ⊢ Rel 𝐿
8	7	a1i 11	. . . . . 6 ⊢ (⊤ → Rel 𝐿)
9		simpr 484	. . . . . . 7 ⊢ ((⊤ ∧ 𝑓𝐿𝑔) → 𝑓𝐿𝑔)
10		eqcom 2743	. . . . . . . . . 10 ⊢ ((𝑎‘0) = (𝑏‘0) ↔ (𝑏‘0) = (𝑎‘0))
11	10	2rexbii 3112	. . . . . . . . 9 ⊢ (∃𝑎 ∈ (◡𝑆 “ {𝑓})∃𝑏 ∈ (◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ (◡𝑆 “ {𝑓})∃𝑏 ∈ (◡𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0))
12		rexcom 3265	. . . . . . . . 9 ⊢ (∃𝑎 ∈ (◡𝑆 “ {𝑓})∃𝑏 ∈ (◡𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0) ↔ ∃𝑏 ∈ (◡𝑆 “ {𝑔})∃𝑎 ∈ (◡𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
13	11, 12	bitri 275	. . . . . . . 8 ⊢ (∃𝑎 ∈ (◡𝑆 “ {𝑓})∃𝑏 ∈ (◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑏 ∈ (◡𝑆 “ {𝑔})∃𝑎 ∈ (◡𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
14		efgred.d	. . . . . . . . 9 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
15		efgred.s	. . . . . . . . 9 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
16	1, 2, 3, 4, 14, 15, 6	efgrelexlema 19678	. . . . . . . 8 ⊢ (𝑓𝐿𝑔 ↔ ∃𝑎 ∈ (◡𝑆 “ {𝑓})∃𝑏 ∈ (◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0))
17	1, 2, 3, 4, 14, 15, 6	efgrelexlema 19678	. . . . . . . 8 ⊢ (𝑔𝐿𝑓 ↔ ∃𝑏 ∈ (◡𝑆 “ {𝑔})∃𝑎 ∈ (◡𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
18	13, 16, 17	3bitr4i 303	. . . . . . 7 ⊢ (𝑓𝐿𝑔 ↔ 𝑔𝐿𝑓)
19	9, 18	sylib 218	. . . . . 6 ⊢ ((⊤ ∧ 𝑓𝐿𝑔) → 𝑔𝐿𝑓)
20	1, 2, 3, 4, 14, 15, 6	efgrelexlema 19678	. . . . . . . . 9 ⊢ (𝑔𝐿ℎ ↔ ∃𝑟 ∈ (◡𝑆 “ {𝑔})∃𝑠 ∈ (◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0))
21		reeanv 3208	. . . . . . . . . 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)))
22	1, 2, 3, 4, 14, 15	efgsfo 19668	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑆:dom 𝑆–onto→𝑊
23		fofn 6748	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆:dom 𝑆–onto→𝑊 → 𝑆 Fn dom 𝑆)
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑆 Fn dom 𝑆
25		fniniseg 7005	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 Fn dom 𝑆 → (𝑟 ∈ (◡𝑆 “ {𝑔}) ↔ (𝑟 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = 𝑔)))
26	24, 25	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ (◡𝑆 “ {𝑔}) ↔ (𝑟 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = 𝑔))
27		fniniseg 7005	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 Fn dom 𝑆 → (𝑏 ∈ (◡𝑆 “ {𝑔}) ↔ (𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑏) = 𝑔)))
28	24, 27	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ (◡𝑆 “ {𝑔}) ↔ (𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑏) = 𝑔))
29		eqtr3 2758	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆‘𝑟) = 𝑔 ∧ (𝑆‘𝑏) = 𝑔) → (𝑆‘𝑟) = (𝑆‘𝑏))
30	1, 2, 3, 4, 14, 15	efgred 19677	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = (𝑆‘𝑏)) → (𝑟‘0) = (𝑏‘0))
31	30	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = (𝑆‘𝑏)) → (𝑏‘0) = (𝑟‘0))
32	31	3expa 1118	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆) ∧ (𝑆‘𝑟) = (𝑆‘𝑏)) → (𝑏‘0) = (𝑟‘0))
33	29, 32	sylan2 593	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆) ∧ ((𝑆‘𝑟) = 𝑔 ∧ (𝑆‘𝑏) = 𝑔)) → (𝑏‘0) = (𝑟‘0))
34	33	an4s 660	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑟 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = 𝑔) ∧ (𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑏) = 𝑔)) → (𝑏‘0) = (𝑟‘0))
35	26, 28, 34	syl2anb 598	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑟 ∈ (◡𝑆 “ {𝑔}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑔})) → (𝑏‘0) = (𝑟‘0))
36		eqeq2 2748	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑟‘0) = (𝑠‘0) → ((𝑏‘0) = (𝑟‘0) ↔ (𝑏‘0) = (𝑠‘0)))
37	35, 36	syl5ibcom 245	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ (◡𝑆 “ {𝑔}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑔})) → ((𝑟‘0) = (𝑠‘0) → (𝑏‘0) = (𝑠‘0)))
38	37	reximdv 3151	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∈ (◡𝑆 “ {𝑔}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑔})) → (∃𝑠 ∈ (◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (◡𝑆 “ {ℎ})(𝑏‘0) = (𝑠‘0)))
39		eqeq1 2740	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎‘0) = (𝑏‘0) → ((𝑎‘0) = (𝑠‘0) ↔ (𝑏‘0) = (𝑠‘0)))
40	39	rexbidv 3160	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0) ↔ ∃𝑠 ∈ (◡𝑆 “ {ℎ})(𝑏‘0) = (𝑠‘0)))
41	40	imbi2d 340	. . . . . . . . . . . . . . 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))))
42	38, 41	syl5ibrcom 247	. . . . . . . . . . . . . 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))))
43	42	rexlimdva 3137	. . . . . . . . . . . . 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))))
44	43	impd 410	. . . . . . . . . . . 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)))
45	44	rexlimiv 3130	. . . . . . . . . . 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))
46	45	reximi 3074	. . . . . . . . . 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))
47	21, 46	sylbir 235	. . . . . . . . 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))
48	16, 20, 47	syl2anb 598	. . . . . . . 8 ⊢ ((𝑓𝐿𝑔 ∧ 𝑔𝐿ℎ) → ∃𝑎 ∈ (◡𝑆 “ {𝑓})∃𝑠 ∈ (◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0))
49	1, 2, 3, 4, 14, 15, 6	efgrelexlema 19678	. . . . . . . 8 ⊢ (𝑓𝐿ℎ ↔ ∃𝑎 ∈ (◡𝑆 “ {𝑓})∃𝑠 ∈ (◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0))
50	48, 49	sylibr 234	. . . . . . 7 ⊢ ((𝑓𝐿𝑔 ∧ 𝑔𝐿ℎ) → 𝑓𝐿ℎ)
51	50	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ (𝑓𝐿𝑔 ∧ 𝑔𝐿ℎ)) → 𝑓𝐿ℎ)
52		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑎‘0) = (𝑎‘0)
53		fveq1 6833	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑎 → (𝑏‘0) = (𝑎‘0))
54	53	rspceeqv 3599	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ (◡𝑆 “ {𝑓}) ∧ (𝑎‘0) = (𝑎‘0)) → ∃𝑏 ∈ (◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
55	52, 54	mpan2 691	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (◡𝑆 “ {𝑓}) → ∃𝑏 ∈ (◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
56	55	pm4.71i 559	. . . . . . . . . 10 ⊢ (𝑎 ∈ (◡𝑆 “ {𝑓}) ↔ (𝑎 ∈ (◡𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)))
57		fniniseg 7005	. . . . . . . . . . 11 ⊢ (𝑆 Fn dom 𝑆 → (𝑎 ∈ (◡𝑆 “ {𝑓}) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆‘𝑎) = 𝑓)))
58	24, 57	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑎 ∈ (◡𝑆 “ {𝑓}) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆‘𝑎) = 𝑓))
59	56, 58	bitr3i 277	. . . . . . . . 9 ⊢ ((𝑎 ∈ (◡𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆‘𝑎) = 𝑓))
60	59	rexbii2 3079	. . . . . . . 8 ⊢ (∃𝑎 ∈ (◡𝑆 “ {𝑓})∃𝑏 ∈ (◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ dom 𝑆(𝑆‘𝑎) = 𝑓)
61	1, 2, 3, 4, 14, 15, 6	efgrelexlema 19678	. . . . . . . 8 ⊢ (𝑓𝐿𝑓 ↔ ∃𝑎 ∈ (◡𝑆 “ {𝑓})∃𝑏 ∈ (◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
62		forn 6749	. . . . . . . . . . 11 ⊢ (𝑆:dom 𝑆–onto→𝑊 → ran 𝑆 = 𝑊)
63	22, 62	ax-mp 5	. . . . . . . . . 10 ⊢ ran 𝑆 = 𝑊
64	63	eleq2i 2828	. . . . . . . . 9 ⊢ (𝑓 ∈ ran 𝑆 ↔ 𝑓 ∈ 𝑊)
65		fvelrnb 6894	. . . . . . . . . 10 ⊢ (𝑆 Fn dom 𝑆 → (𝑓 ∈ ran 𝑆 ↔ ∃𝑎 ∈ dom 𝑆(𝑆‘𝑎) = 𝑓))
66	24, 65	ax-mp 5	. . . . . . . . 9 ⊢ (𝑓 ∈ ran 𝑆 ↔ ∃𝑎 ∈ dom 𝑆(𝑆‘𝑎) = 𝑓)
67	64, 66	bitr3i 277	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑊 ↔ ∃𝑎 ∈ dom 𝑆(𝑆‘𝑎) = 𝑓)
68	60, 61, 67	3bitr4ri 304	. . . . . . 7 ⊢ (𝑓 ∈ 𝑊 ↔ 𝑓𝐿𝑓)
69	68	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑓 ∈ 𝑊 ↔ 𝑓𝐿𝑓))
70	8, 19, 51, 69	iserd 8661	. . . . 5 ⊢ (⊤ → 𝐿 Er 𝑊)
71	70	mptru 1548	. . . 4 ⊢ 𝐿 Er 𝑊
72		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑎 ∈ 𝑊)
73		foelrn 7052	. . . . . . . . . . 11 ⊢ ((𝑆:dom 𝑆–onto→𝑊 ∧ 𝑎 ∈ 𝑊) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆‘𝑟))
74	22, 72, 73	sylancr 587	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆‘𝑟))
75		simprl 770	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ dom 𝑆)
76		simprr 772	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑎 = (𝑆‘𝑟))
77	76	eqcomd 2742	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑆‘𝑟) = 𝑎)
78		fniniseg 7005	. . . . . . . . . . . 12 ⊢ (𝑆 Fn dom 𝑆 → (𝑟 ∈ (◡𝑆 “ {𝑎}) ↔ (𝑟 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = 𝑎)))
79	24, 78	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑟 ∈ (◡𝑆 “ {𝑎}) ↔ (𝑟 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = 𝑎))
80	75, 77, 79	sylanbrc 583	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ (◡𝑆 “ {𝑎}))
81		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑏 ∈ ran (𝑇‘𝑎))
82	76	fveq2d 6838	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑇‘𝑎) = (𝑇‘(𝑆‘𝑟)))
83	82	rneqd 5887	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → ran (𝑇‘𝑎) = ran (𝑇‘(𝑆‘𝑟)))
84	81, 83	eleqtrd 2838	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑏 ∈ ran (𝑇‘(𝑆‘𝑟)))
85	1, 2, 3, 4, 14, 15	efgsp1 19666	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ ran (𝑇‘(𝑆‘𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆)
86	75, 84, 85	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆)
87	1, 2, 3, 4, 14, 15	efgsdm 19659	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ dom 𝑆 ↔ (𝑟 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑟‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑟))(𝑟‘𝑖) ∈ ran (𝑇‘(𝑟‘(𝑖 − 1)))))
88	87	simp1bi 1145	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ dom 𝑆 → 𝑟 ∈ (Word 𝑊 ∖ {∅}))
89	88	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ (Word 𝑊 ∖ {∅}))
90	89	eldifad 3913	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ Word 𝑊)
91	1, 2, 3, 4	efgtf 19651	. . . . . . . . . . . . . . . . 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))
92	91	simprd 495	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝑊 → (𝑇‘𝑎):((0...(♯‘𝑎)) × (𝐼 × 2o))⟶𝑊)
93	92	frnd 6670	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝑊 → ran (𝑇‘𝑎) ⊆ 𝑊)
94	93	sselda 3933	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑏 ∈ 𝑊)
95	94	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑏 ∈ 𝑊)
96	1, 2, 3, 4, 14, 15	efgsval2 19662	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ Word 𝑊 ∧ 𝑏 ∈ 𝑊 ∧ (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
97	90, 95, 86, 96	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
98		fniniseg 7005	. . . . . . . . . . . . 13 ⊢ (𝑆 Fn dom 𝑆 → ((𝑟 ++ ⟨“𝑏”⟩) ∈ (◡𝑆 “ {𝑏}) ↔ ((𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆 ∧ (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)))
99	24, 98	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑟 ++ ⟨“𝑏”⟩) ∈ (◡𝑆 “ {𝑏}) ↔ ((𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆 ∧ (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏))
100	86, 97, 99	sylanbrc 583	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ (◡𝑆 “ {𝑏}))
101	95	s1cld 14527	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → ⟨“𝑏”⟩ ∈ Word 𝑊)
102		eldifsn 4742	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ (Word 𝑊 ∖ {∅}) ↔ (𝑟 ∈ Word 𝑊 ∧ 𝑟 ≠ ∅))
103		lennncl 14457	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ Word 𝑊 ∧ 𝑟 ≠ ∅) → (♯‘𝑟) ∈ ℕ)
104	102, 103	sylbi 217	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ (Word 𝑊 ∖ {∅}) → (♯‘𝑟) ∈ ℕ)
105	89, 104	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (♯‘𝑟) ∈ ℕ)
106		lbfzo0 13615	. . . . . . . . . . . . . 14 ⊢ (0 ∈ (0..^(♯‘𝑟)) ↔ (♯‘𝑟) ∈ ℕ)
107	105, 106	sylibr 234	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 0 ∈ (0..^(♯‘𝑟)))
108		ccatval1 14500	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ Word 𝑊 ∧ ⟨“𝑏”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(♯‘𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
109	90, 101, 107, 108	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
110	109	eqcomd 2742	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
111		fveq1 6833	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑟 ++ ⟨“𝑏”⟩) → (𝑠‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
112	111	rspceeqv 3599	. . . . . . . . . . 11 ⊢ (((𝑟 ++ ⟨“𝑏”⟩) ∈ (◡𝑆 “ {𝑏}) ∧ (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0)) → ∃𝑠 ∈ (◡𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
113	100, 110, 112	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → ∃𝑠 ∈ (◡𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
114	74, 80, 113	reximssdv 3154	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → ∃𝑟 ∈ (◡𝑆 “ {𝑎})∃𝑠 ∈ (◡𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
115	1, 2, 3, 4, 14, 15, 6	efgrelexlema 19678	. . . . . . . . 9 ⊢ (𝑎𝐿𝑏 ↔ ∃𝑟 ∈ (◡𝑆 “ {𝑎})∃𝑠 ∈ (◡𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
116	114, 115	sylibr 234	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑎𝐿𝑏)
117		vex 3444	. . . . . . . . 9 ⊢ 𝑏 ∈ V
118		vex 3444	. . . . . . . . 9 ⊢ 𝑎 ∈ V
119	117, 118	elec 8681	. . . . . . . 8 ⊢ (𝑏 ∈ [𝑎]𝐿 ↔ 𝑎𝐿𝑏)
120	116, 119	sylibr 234	. . . . . . 7 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑏 ∈ [𝑎]𝐿)
121	120	ex 412	. . . . . 6 ⊢ (𝑎 ∈ 𝑊 → (𝑏 ∈ ran (𝑇‘𝑎) → 𝑏 ∈ [𝑎]𝐿))
122	121	ssrdv 3939	. . . . 5 ⊢ (𝑎 ∈ 𝑊 → ran (𝑇‘𝑎) ⊆ [𝑎]𝐿)
123	122	rgen 3053	. . . 4 ⊢ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝐿
124	1	fvexi 6848	. . . . . 6 ⊢ 𝑊 ∈ V
125		erex 8659	. . . . . 6 ⊢ (𝐿 Er 𝑊 → (𝑊 ∈ V → 𝐿 ∈ V))
126	71, 124, 125	mp2 9	. . . . 5 ⊢ 𝐿 ∈ V
127		ereq1 8642	. . . . . 6 ⊢ (𝑟 = 𝐿 → (𝑟 Er 𝑊 ↔ 𝐿 Er 𝑊))
128		eceq2 8676	. . . . . . . 8 ⊢ (𝑟 = 𝐿 → [𝑎]𝑟 = [𝑎]𝐿)
129	128	sseq2d 3966	. . . . . . 7 ⊢ (𝑟 = 𝐿 → (ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 ↔ ran (𝑇‘𝑎) ⊆ [𝑎]𝐿))
130	129	ralbidv 3159	. . . . . 6 ⊢ (𝑟 = 𝐿 → (∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 ↔ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝐿))
131	127, 130	anbi12d 632	. . . . 5 ⊢ (𝑟 = 𝐿 → ((𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟) ↔ (𝐿 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝐿)))
132	126, 131	elab 3634	. . . 4 ⊢ (𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟)} ↔ (𝐿 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝐿))
133	71, 123, 132	mpbir2an 711	. . 3 ⊢ 𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟)}
134		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)
135	133, 134	ax-mp 5	. 2 ⊢ ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟)} ⊆ 𝐿
136	5, 135	eqsstri 3980	1 ⊢ ∼ ⊆ 𝐿

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ⊤wtru 1542   ∈ wcel 2113  {cab 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3399  Vcvv 3440   ∖ cdif 3898   ⊆ wss 3901  ∅c0 4285  {csn 4580  ⟨cop 4586  ⟨cotp 4588  ∩ cint 4902  ∪ ciun 4946   class class class wbr 5098  {copab 5160   ↦ cmpt 5179   I cid 5518   × cxp 5622  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   “ cima 5627  Rel wrel 5629   Fn wfn 6487  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  1_oc1o 8390  2_oc2o 8391   Er wer 8632  [cec 8633  0cc0 11026  1c1 11027   − cmin 11364  ℕcn 12145  ...cfz 13423  ..^cfzo 13570  ♯chash 14253  Word cword 14436   ++ cconcat 14493  ⟨“cs1 14519   splice csplice 14672  ⟨“cs2 14764   ~_FG cefg 19635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-ot 4589  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-ec 8637  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-n0 12402  df-xnn0 12475  df-z 12489  df-uz 12752  df-rp 12906  df-fz 13424  df-fzo 13571  df-hash 14254  df-word 14437  df-concat 14494  df-s1 14520  df-substr 14565  df-pfx 14595  df-splice 14673  df-s2 14771  df-efg 19638

This theorem is referenced by:  efgrelex  19680