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Theorem efgrelexlemb 18443

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 18415	. 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	relopabi 5416	. . . . . . 7 ⊢ Rel 𝐿
8	7	a1i 11	. . . . . 6 ⊢ (⊤ → Rel 𝐿)
9		simpr 477	. . . . . . 7 ⊢ ((⊤ ∧ 𝑓𝐿𝑔) → 𝑓𝐿𝑔)
10		eqcom 2772	. . . . . . . . . 10 ⊢ ((𝑎‘0) = (𝑏‘0) ↔ (𝑏‘0) = (𝑎‘0))
11	10	2rexbii 3189	. . . . . . . . 9 ⊢ (∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑏‘0) = (𝑎‘0))
12		rexcom 3246	. . . . . . . . 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 266	. . . . . . . 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 18442	. . . . . . . 8 ⊢ (𝑓𝐿𝑔 ↔ ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑔})(𝑎‘0) = (𝑏‘0))
17	1, 2, 3, 4, 14, 15, 6	efgrelexlema 18442	. . . . . . . 8 ⊢ (𝑔𝐿𝑓 ↔ ∃𝑏 ∈ (^◡𝑆 “ {𝑔})∃𝑎 ∈ (^◡𝑆 “ {𝑓})(𝑏‘0) = (𝑎‘0))
18	13, 16, 17	3bitr4i 294	. . . . . . 7 ⊢ (𝑓𝐿𝑔 ↔ 𝑔𝐿𝑓)
19	9, 18	sylib 209	. . . . . 6 ⊢ ((⊤ ∧ 𝑓𝐿𝑔) → 𝑔𝐿𝑓)
20	1, 2, 3, 4, 14, 15, 6	efgrelexlema 18442	. . . . . . . . 9 ⊢ (𝑔𝐿ℎ ↔ ∃𝑟 ∈ (^◡𝑆 “ {𝑔})∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0))
21		reeanv 3254	. . . . . . . . . 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 18431	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑆:dom 𝑆–onto→𝑊
23		fofn 6302	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆:dom 𝑆–onto→𝑊 → 𝑆 Fn dom 𝑆)
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑆 Fn dom 𝑆
25		fniniseg 6532	. . . . . . . . . . . . . . . . . . 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 6532	. . . . . . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆‘𝑟) = 𝑔 ∧ (𝑆‘𝑏) = 𝑔) → (𝑆‘𝑟) = (𝑆‘𝑏))
30	1, 2, 3, 4, 14, 15	efgred 18441	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = (𝑆‘𝑏)) → (𝑟‘0) = (𝑏‘0))
31	30	eqcomd 2771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = (𝑆‘𝑏)) → (𝑏‘0) = (𝑟‘0))
32	31	3expa 1147	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆) ∧ (𝑆‘𝑟) = (𝑆‘𝑏)) → (𝑏‘0) = (𝑟‘0))
33	29, 32	sylan2 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆) ∧ ((𝑆‘𝑟) = 𝑔 ∧ (𝑆‘𝑏) = 𝑔)) → (𝑏‘0) = (𝑟‘0))
34	33	an4s 650	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑟 ∈ dom 𝑆 ∧ (𝑆‘𝑟) = 𝑔) ∧ (𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑏) = 𝑔)) → (𝑏‘0) = (𝑟‘0))
35	26, 28, 34	syl2anb 591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑟 ∈ (^◡𝑆 “ {𝑔}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑔})) → (𝑏‘0) = (𝑟‘0))
36		eqeq2 2776	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑟‘0) = (𝑠‘0) → ((𝑏‘0) = (𝑟‘0) ↔ (𝑏‘0) = (𝑠‘0)))
37	35, 36	syl5ibcom 236	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ (^◡𝑆 “ {𝑔}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑔})) → ((𝑟‘0) = (𝑠‘0) → (𝑏‘0) = (𝑠‘0)))
38	37	reximdv 3162	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∈ (^◡𝑆 “ {𝑔}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑔})) → (∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑟‘0) = (𝑠‘0) → ∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑏‘0) = (𝑠‘0)))
39		eqeq1 2769	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎‘0) = (𝑏‘0) → ((𝑎‘0) = (𝑠‘0) ↔ (𝑏‘0) = (𝑠‘0)))
40	39	rexbidv 3199	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎‘0) = (𝑏‘0) → (∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0) ↔ ∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑏‘0) = (𝑠‘0)))
41	40	imbi2d 331	. . . . . . . . . . . . . . 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 238	. . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . 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 398	. . . . . . . . . . . 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 3174	. . . . . . . . . . 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 3157	. . . . . . . . . 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 226	. . . . . . . . 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 591	. . . . . . . 8 ⊢ ((𝑓𝐿𝑔 ∧ 𝑔𝐿ℎ) → ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0))
49	1, 2, 3, 4, 14, 15, 6	efgrelexlema 18442	. . . . . . . 8 ⊢ (𝑓𝐿ℎ ↔ ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑠 ∈ (^◡𝑆 “ {ℎ})(𝑎‘0) = (𝑠‘0))
50	48, 49	sylibr 225	. . . . . . 7 ⊢ ((𝑓𝐿𝑔 ∧ 𝑔𝐿ℎ) → 𝑓𝐿ℎ)
51	50	adantl 473	. . . . . 6 ⊢ ((⊤ ∧ (𝑓𝐿𝑔 ∧ 𝑔𝐿ℎ)) → 𝑓𝐿ℎ)
52		eqid 2765	. . . . . . . . . . . 12 ⊢ (𝑎‘0) = (𝑎‘0)
53		fveq1 6378	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑎 → (𝑏‘0) = (𝑎‘0))
54	53	rspceeqv 3480	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ (^◡𝑆 “ {𝑓}) ∧ (𝑎‘0) = (𝑎‘0)) → ∃𝑏 ∈ (^◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
55	52, 54	mpan2 682	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (^◡𝑆 “ {𝑓}) → ∃𝑏 ∈ (^◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
56	55	pm4.71i 555	. . . . . . . . . 10 ⊢ (𝑎 ∈ (^◡𝑆 “ {𝑓}) ↔ (𝑎 ∈ (^◡𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (^◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)))
57		fniniseg 6532	. . . . . . . . . . 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 268	. . . . . . . . 9 ⊢ ((𝑎 ∈ (^◡𝑆 “ {𝑓}) ∧ ∃𝑏 ∈ (^◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0)) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆‘𝑎) = 𝑓))
60	59	rexbii2 3186	. . . . . . . 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 18442	. . . . . . . 8 ⊢ (𝑓𝐿𝑓 ↔ ∃𝑎 ∈ (^◡𝑆 “ {𝑓})∃𝑏 ∈ (^◡𝑆 “ {𝑓})(𝑎‘0) = (𝑏‘0))
62		forn 6303	. . . . . . . . . . 11 ⊢ (𝑆:dom 𝑆–onto→𝑊 → ran 𝑆 = 𝑊)
63	22, 62	ax-mp 5	. . . . . . . . . 10 ⊢ ran 𝑆 = 𝑊
64	63	eleq2i 2836	. . . . . . . . 9 ⊢ (𝑓 ∈ ran 𝑆 ↔ 𝑓 ∈ 𝑊)
65		fvelrnb 6436	. . . . . . . . . 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 268	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑊 ↔ ∃𝑎 ∈ dom 𝑆(𝑆‘𝑎) = 𝑓)
68	60, 61, 67	3bitr4ri 295	. . . . . . 7 ⊢ (𝑓 ∈ 𝑊 ↔ 𝑓𝐿𝑓)
69	68	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑓 ∈ 𝑊 ↔ 𝑓𝐿𝑓))
70	8, 19, 51, 69	iserd 7977	. . . . 5 ⊢ (⊤ → 𝐿 Er 𝑊)
71	70	mptru 1660	. . . 4 ⊢ 𝐿 Er 𝑊
72		simpl 474	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑎 ∈ 𝑊)
73		foelrn 6572	. . . . . . . . . . 11 ⊢ ((𝑆:dom 𝑆–onto→𝑊 ∧ 𝑎 ∈ 𝑊) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆‘𝑟))
74	22, 72, 73	sylancr 581	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → ∃𝑟 ∈ dom 𝑆 𝑎 = (𝑆‘𝑟))
75		simprl 787	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ dom 𝑆)
76		simprr 789	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑎 = (𝑆‘𝑟))
77	76	eqcomd 2771	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑆‘𝑟) = 𝑎)
78		fniniseg 6532	. . . . . . . . . . . 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 578	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ (^◡𝑆 “ {𝑎}))
81		simplr 785	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑏 ∈ ran (𝑇‘𝑎))
82	76	fveq2d 6383	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑇‘𝑎) = (𝑇‘(𝑆‘𝑟)))
83	82	rneqd 5523	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → ran (𝑇‘𝑎) = ran (𝑇‘(𝑆‘𝑟)))
84	81, 83	eleqtrd 2846	. . . . . . . . . . . . 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 18428	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ dom 𝑆 ∧ 𝑏 ∈ ran (𝑇‘(𝑆‘𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆)
86	75, 84, 85	syl2anc 579	. . . . . . . . . . . 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 18421	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ dom 𝑆 ↔ (𝑟 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑟‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑟))(𝑟‘𝑖) ∈ ran (𝑇‘(𝑟‘(𝑖 − 1)))))
88	87	simp1bi 1175	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ dom 𝑆 → 𝑟 ∈ (Word 𝑊 ∖ {∅}))
89	88	ad2antrl 719	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ (Word 𝑊 ∖ {∅}))
90	89	eldifad 3746	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑟 ∈ Word 𝑊)
91	1, 2, 3, 4	efgtf 18413	. . . . . . . . . . . . . . . . 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))
92	91	simprd 489	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝑊 → (𝑇‘𝑎):((0...(♯‘𝑎)) × (𝐼 × 2_𝑜))⟶𝑊)
93	92	frnd 6232	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝑊 → ran (𝑇‘𝑎) ⊆ 𝑊)
94	93	sselda 3763	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑏 ∈ 𝑊)
95	94	adantr 472	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 𝑏 ∈ 𝑊)
96	1, 2, 3, 4, 14, 15	efgsval2 18424	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ Word 𝑊 ∧ 𝑏 ∈ 𝑊 ∧ (𝑟 ++ ⟨“𝑏”⟩) ∈ dom 𝑆) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
97	90, 95, 86, 96	syl3anc 1490	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑆‘(𝑟 ++ ⟨“𝑏”⟩)) = 𝑏)
98		fniniseg 6532	. . . . . . . . . . . . 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 578	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑟 ++ ⟨“𝑏”⟩) ∈ (^◡𝑆 “ {𝑏}))
101	95	s1cld 13580	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → ⟨“𝑏”⟩ ∈ Word 𝑊)
102		eldifsn 4474	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ (Word 𝑊 ∖ {∅}) ↔ (𝑟 ∈ Word 𝑊 ∧ 𝑟 ≠ ∅))
103		lennncl 13511	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ Word 𝑊 ∧ 𝑟 ≠ ∅) → (♯‘𝑟) ∈ ℕ)
104	102, 103	sylbi 208	. . . . . . . . . . . . . . 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 12721	. . . . . . . . . . . . . 14 ⊢ (0 ∈ (0..^(♯‘𝑟)) ↔ (♯‘𝑟) ∈ ℕ)
107	105, 106	sylibr 225	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → 0 ∈ (0..^(♯‘𝑟)))
108		ccatval1 13554	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ Word 𝑊 ∧ ⟨“𝑏”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(♯‘𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
109	90, 101, 107, 108	syl3anc 1490	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → ((𝑟 ++ ⟨“𝑏”⟩)‘0) = (𝑟‘0))
110	109	eqcomd 2771	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) ∧ (𝑟 ∈ dom 𝑆 ∧ 𝑎 = (𝑆‘𝑟))) → (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
111		fveq1 6378	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑟 ++ ⟨“𝑏”⟩) → (𝑠‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0))
112	111	rspceeqv 3480	. . . . . . . . . . 11 ⊢ (((𝑟 ++ ⟨“𝑏”⟩) ∈ (^◡𝑆 “ {𝑏}) ∧ (𝑟‘0) = ((𝑟 ++ ⟨“𝑏”⟩)‘0)) → ∃𝑠 ∈ (^◡𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
113	100, 110, 112	syl2anc 579	. . . . . . . . . 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 3165	. . . . . . . . 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 18442	. . . . . . . . 9 ⊢ (𝑎𝐿𝑏 ↔ ∃𝑟 ∈ (^◡𝑆 “ {𝑎})∃𝑠 ∈ (^◡𝑆 “ {𝑏})(𝑟‘0) = (𝑠‘0))
116	114, 115	sylibr 225	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑎𝐿𝑏)
117		vex 3353	. . . . . . . . 9 ⊢ 𝑏 ∈ V
118		vex 3353	. . . . . . . . 9 ⊢ 𝑎 ∈ V
119	117, 118	elec 7993	. . . . . . . 8 ⊢ (𝑏 ∈ [𝑎]𝐿 ↔ 𝑎𝐿𝑏)
120	116, 119	sylibr 225	. . . . . . 7 ⊢ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ ran (𝑇‘𝑎)) → 𝑏 ∈ [𝑎]𝐿)
121	120	ex 401	. . . . . 6 ⊢ (𝑎 ∈ 𝑊 → (𝑏 ∈ ran (𝑇‘𝑎) → 𝑏 ∈ [𝑎]𝐿))
122	121	ssrdv 3769	. . . . 5 ⊢ (𝑎 ∈ 𝑊 → ran (𝑇‘𝑎) ⊆ [𝑎]𝐿)
123	122	rgen 3069	. . . 4 ⊢ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝐿
124	1	fvexi 6393	. . . . . 6 ⊢ 𝑊 ∈ V
125		erex 7975	. . . . . 6 ⊢ (𝐿 Er 𝑊 → (𝑊 ∈ V → 𝐿 ∈ V))
126	71, 124, 125	mp2 9	. . . . 5 ⊢ 𝐿 ∈ V
127		ereq1 7958	. . . . . 6 ⊢ (𝑟 = 𝐿 → (𝑟 Er 𝑊 ↔ 𝐿 Er 𝑊))
128		eceq2 7991	. . . . . . . 8 ⊢ (𝑟 = 𝐿 → [𝑎]𝑟 = [𝑎]𝐿)
129	128	sseq2d 3795	. . . . . . 7 ⊢ (𝑟 = 𝐿 → (ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 ↔ ran (𝑇‘𝑎) ⊆ [𝑎]𝐿))
130	129	ralbidv 3133	. . . . . 6 ⊢ (𝑟 = 𝐿 → (∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 ↔ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝐿))
131	127, 130	anbi12d 624	. . . . 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 3507	. . . 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 702	. . 3 ⊢ 𝐿 ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟)}
134		intss1 4650	. . 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 3797	1 ⊢ ∼ ⊆ 𝐿

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 197 ∧ wa 384 ∧ w3a 1107 = wceq 1652 ⊤wtru 1653 ∈ wcel 2155 {cab 2751 ≠ wne 2937 ∀wral 3055 ∃wrex 3056 {crab 3059 Vcvv 3350 ∖ cdif 3731 ⊆ wss 3734 ∅c0 4081 {csn 4336 ⟨cop 4342 ⟨cotp 4344 ∩ cint 4635 ∪ ciun 4678 class class class wbr 4811 {copab 4873 ↦ cmpt 4890 I cid 5186 × cxp 5277 ^◡ccnv 5278 dom cdm 5279 ran crn 5280 “ cima 5282 Rel wrel 5284 Fn wfn 6065 ⟶wf 6066 –onto→wfo 6068 ‘cfv 6070 (class class class)co 6846 ↦ cmpt2 6848 1_𝑜c1o 7761 2_𝑜c2o 7762 Er wer 7948 [cec 7949 0cc0 10193 1c1 10194 − cmin 10524 ℕcn 11278 ...cfz 12538 ..^cfzo 12678 ♯chash 13326 Word cword 13491 ++ cconcat 13547 ⟨“cs1 13572 splice csplice 13777 ⟨“cs2 13884 ~_FG cefg 18397

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1890 ax-4 1904 ax-5 2005 ax-6 2070 ax-7 2105 ax-8 2157 ax-9 2164 ax-10 2183 ax-11 2198 ax-12 2211 ax-13 2352 ax-ext 2743 ax-rep 4932 ax-sep 4943 ax-nul 4951 ax-pow 5003 ax-pr 5064 ax-un 7151 ax-cnex 10249 ax-resscn 10250 ax-1cn 10251 ax-icn 10252 ax-addcl 10253 ax-addrcl 10254 ax-mulcl 10255 ax-mulrcl 10256 ax-mulcom 10257 ax-addass 10258 ax-mulass 10259 ax-distr 10260 ax-i2m1 10261 ax-1ne0 10262 ax-1rid 10263 ax-rnegex 10264 ax-rrecex 10265 ax-cnre 10266 ax-pre-lttri 10267 ax-pre-lttrn 10268 ax-pre-ltadd 10269 ax-pre-mulgt0 10270

This theorem depends on definitions: df-bi 198 df-an 385 df-or 874 df-3or 1108 df-3an 1109 df-tru 1656 df-ex 1875 df-nf 1879 df-sb 2063 df-mo 2565 df-eu 2582 df-clab 2752 df-cleq 2758 df-clel 2761 df-nfc 2896 df-ne 2938 df-nel 3041 df-ral 3060 df-rex 3061 df-reu 3062 df-rab 3064 df-v 3352 df-sbc 3599 df-csb 3694 df-dif 3737 df-un 3739 df-in 3741 df-ss 3748 df-pss 3750 df-nul 4082 df-if 4246 df-pw 4319 df-sn 4337 df-pr 4339 df-tp 4341 df-op 4343 df-ot 4345 df-uni 4597 df-int 4636 df-iun 4680 df-br 4812 df-opab 4874 df-mpt 4891 df-tr 4914 df-id 5187 df-eprel 5192 df-po 5200 df-so 5201 df-fr 5238 df-we 5240 df-xp 5285 df-rel 5286 df-cnv 5287 df-co 5288 df-dm 5289 df-rn 5290 df-res 5291 df-ima 5292 df-pred 5867 df-ord 5913 df-on 5914 df-lim 5915 df-suc 5916 df-iota 6033 df-fun 6072 df-fn 6073 df-f 6074 df-f1 6075 df-fo 6076 df-f1o 6077 df-fv 6078 df-riota 6807 df-ov 6849 df-oprab 6850 df-mpt2 6851 df-om 7268 df-1st 7370 df-2nd 7371 df-wrecs 7614 df-recs 7676 df-rdg 7714 df-1o 7768 df-2o 7769 df-oadd 7772 df-er 7951 df-ec 7953 df-map 8066 df-pm 8067 df-en 8165 df-dom 8166 df-sdom 8167 df-fin 8168 df-card 9020 df-pnf 10334 df-mnf 10335 df-xr 10336 df-ltxr 10337 df-le 10338 df-sub 10526 df-neg 10527 df-nn 11279 df-2 11339 df-n0 11543 df-z 11629 df-uz 11892 df-rp 12034 df-fz 12539 df-fzo 12679 df-hash 13327 df-word 13492 df-concat 13548 df-s1 13573 df-substr 13623 df-pfx 13672 df-splice 13779 df-s2 13891 df-efg 18400

This theorem is referenced by: efgrelex 18444

W3C validator