Theorem efgredeu 19785

Description: There is a unique reduced word equivalent to a given word. (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)))

Assertion

Ref	Expression
efgredeu	⊢ (𝐴 ∈ 𝑊 → ∃!𝑑 ∈ 𝐷 𝑑 ∼ 𝐴)

Distinct variable groups:   𝐴,𝑑   𝑦,𝑧   𝑡,𝑛,𝑣,𝑤,𝑦,𝑧,𝑚,𝑥   𝑚,𝑀   𝑥,𝑛,𝑀,𝑡,𝑣,𝑤   𝑘,𝑚,𝑡,𝑥,𝑇   𝑘,𝑑,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧,𝑊   ∼ ,𝑑,𝑚,𝑡,𝑥,𝑦,𝑧   𝑆,𝑑   𝑚,𝐼,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧   𝐷,𝑑,𝑚,𝑡

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛)   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)   𝑀(𝑦,𝑧,𝑘,𝑑)

Proof of Theorem efgredeu

Dummy variables 𝑎 𝑏 𝑐 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . . . 5 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
2		efgval.r	. . . . 5 ⊢ ∼ = ( ~FG ‘𝐼)
3		efgval2.m	. . . . 5 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
4		efgval2.t	. . . . 5 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
5		efgred.d	. . . . 5 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
6		efgred.s	. . . . 5 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
7	1, 2, 3, 4, 5, 6	efgsfo 19772	. . . 4 ⊢ 𝑆:dom 𝑆–onto→𝑊
8		foelrn 7127	. . . 4 ⊢ ((𝑆:dom 𝑆–onto→𝑊 ∧ 𝐴 ∈ 𝑊) → ∃𝑎 ∈ dom 𝑆 𝐴 = (𝑆‘𝑎))
9	7, 8	mpan 690	. . 3 ⊢ (𝐴 ∈ 𝑊 → ∃𝑎 ∈ dom 𝑆 𝐴 = (𝑆‘𝑎))
10	1, 2, 3, 4, 5, 6	efgsdm 19763	. . . . . . 7 ⊢ (𝑎 ∈ dom 𝑆 ↔ (𝑎 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑎‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑎))(𝑎‘𝑖) ∈ ran (𝑇‘(𝑎‘(𝑖 − 1)))))
11	10	simp2bi 1145	. . . . . 6 ⊢ (𝑎 ∈ dom 𝑆 → (𝑎‘0) ∈ 𝐷)
12	1, 2, 3, 4, 5, 6	efgsrel 19767	. . . . . . 7 ⊢ (𝑎 ∈ dom 𝑆 → (𝑎‘0) ∼ (𝑆‘𝑎))
13	12	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆) → (𝑎‘0) ∼ (𝑆‘𝑎))
14		breq1 5151	. . . . . . 7 ⊢ (𝑑 = (𝑎‘0) → (𝑑 ∼ (𝑆‘𝑎) ↔ (𝑎‘0) ∼ (𝑆‘𝑎)))
15	14	rspcev 3622	. . . . . 6 ⊢ (((𝑎‘0) ∈ 𝐷 ∧ (𝑎‘0) ∼ (𝑆‘𝑎)) → ∃𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝑎))
16	11, 13, 15	syl2an2 686	. . . . 5 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆) → ∃𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝑎))
17		breq2 5152	. . . . . 6 ⊢ (𝐴 = (𝑆‘𝑎) → (𝑑 ∼ 𝐴 ↔ 𝑑 ∼ (𝑆‘𝑎)))
18	17	rexbidv 3177	. . . . 5 ⊢ (𝐴 = (𝑆‘𝑎) → (∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ↔ ∃𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝑎)))
19	16, 18	syl5ibrcom 247	. . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆) → (𝐴 = (𝑆‘𝑎) → ∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴))
20	19	rexlimdva 3153	. . 3 ⊢ (𝐴 ∈ 𝑊 → (∃𝑎 ∈ dom 𝑆 𝐴 = (𝑆‘𝑎) → ∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴))
21	9, 20	mpd 15	. 2 ⊢ (𝐴 ∈ 𝑊 → ∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴)
22	1, 2	efger 19751	. . . . . . 7 ⊢ ∼ Er 𝑊
23	22	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → ∼ Er 𝑊)
24		simprl 771	. . . . . 6 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → 𝑑 ∼ 𝐴)
25		simprr 773	. . . . . 6 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → 𝑐 ∼ 𝐴)
26	23, 24, 25	ertr4d 8763	. . . . 5 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → 𝑑 ∼ 𝑐)
27	1, 2, 3, 4, 5, 6	efgrelex 19784	. . . . . 6 ⊢ (𝑑 ∼ 𝑐 → ∃𝑎 ∈ (◡𝑆 “ {𝑑})∃𝑏 ∈ (◡𝑆 “ {𝑐})(𝑎‘0) = (𝑏‘0))
28		fofn 6823	. . . . . . . . . . . . . 14 ⊢ (𝑆:dom 𝑆–onto→𝑊 → 𝑆 Fn dom 𝑆)
29		fniniseg 7080	. . . . . . . . . . . . . 14 ⊢ (𝑆 Fn dom 𝑆 → (𝑎 ∈ (◡𝑆 “ {𝑑}) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆‘𝑎) = 𝑑)))
30	7, 28, 29	mp2b 10	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (◡𝑆 “ {𝑑}) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆‘𝑎) = 𝑑))
31	30	simplbi 497	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (◡𝑆 “ {𝑑}) → 𝑎 ∈ dom 𝑆)
32	31	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → 𝑎 ∈ dom 𝑆)
33	1, 2, 3, 4, 5, 6	efgsval 19764	. . . . . . . . . . 11 ⊢ (𝑎 ∈ dom 𝑆 → (𝑆‘𝑎) = (𝑎‘((♯‘𝑎) − 1)))
34	32, 33	syl 17	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑆‘𝑎) = (𝑎‘((♯‘𝑎) − 1)))
35	30	simprbi 496	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (◡𝑆 “ {𝑑}) → (𝑆‘𝑎) = 𝑑)
36	35	ad2antrl 728	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑆‘𝑎) = 𝑑)
37		simpllr 776	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷))
38	37	simpld 494	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → 𝑑 ∈ 𝐷)
39	36, 38	eqeltrd 2839	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑆‘𝑎) ∈ 𝐷)
40	1, 2, 3, 4, 5, 6	efgs1b 19769	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ dom 𝑆 → ((𝑆‘𝑎) ∈ 𝐷 ↔ (♯‘𝑎) = 1))
41	32, 40	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → ((𝑆‘𝑎) ∈ 𝐷 ↔ (♯‘𝑎) = 1))
42	39, 41	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (♯‘𝑎) = 1)
43	42	oveq1d 7446	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → ((♯‘𝑎) − 1) = (1 − 1))
44		1m1e0 12336	. . . . . . . . . . . 12 ⊢ (1 − 1) = 0
45	43, 44	eqtrdi 2791	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → ((♯‘𝑎) − 1) = 0)
46	45	fveq2d 6911	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑎‘((♯‘𝑎) − 1)) = (𝑎‘0))
47	34, 36, 46	3eqtr3rd 2784	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑎‘0) = 𝑑)
48		fniniseg 7080	. . . . . . . . . . . . . 14 ⊢ (𝑆 Fn dom 𝑆 → (𝑏 ∈ (◡𝑆 “ {𝑐}) ↔ (𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑏) = 𝑐)))
49	7, 28, 48	mp2b 10	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (◡𝑆 “ {𝑐}) ↔ (𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑏) = 𝑐))
50	49	simplbi 497	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (◡𝑆 “ {𝑐}) → 𝑏 ∈ dom 𝑆)
51	50	ad2antll 729	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → 𝑏 ∈ dom 𝑆)
52	1, 2, 3, 4, 5, 6	efgsval 19764	. . . . . . . . . . 11 ⊢ (𝑏 ∈ dom 𝑆 → (𝑆‘𝑏) = (𝑏‘((♯‘𝑏) − 1)))
53	51, 52	syl 17	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑆‘𝑏) = (𝑏‘((♯‘𝑏) − 1)))
54	49	simprbi 496	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (◡𝑆 “ {𝑐}) → (𝑆‘𝑏) = 𝑐)
55	54	ad2antll 729	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑆‘𝑏) = 𝑐)
56	37	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → 𝑐 ∈ 𝐷)
57	55, 56	eqeltrd 2839	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑆‘𝑏) ∈ 𝐷)
58	1, 2, 3, 4, 5, 6	efgs1b 19769	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ dom 𝑆 → ((𝑆‘𝑏) ∈ 𝐷 ↔ (♯‘𝑏) = 1))
59	51, 58	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → ((𝑆‘𝑏) ∈ 𝐷 ↔ (♯‘𝑏) = 1))
60	57, 59	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (♯‘𝑏) = 1)
61	60	oveq1d 7446	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → ((♯‘𝑏) − 1) = (1 − 1))
62	61, 44	eqtrdi 2791	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → ((♯‘𝑏) − 1) = 0)
63	62	fveq2d 6911	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑏‘((♯‘𝑏) − 1)) = (𝑏‘0))
64	53, 55, 63	3eqtr3rd 2784	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑏‘0) = 𝑐)
65	47, 64	eqeq12d 2751	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → ((𝑎‘0) = (𝑏‘0) ↔ 𝑑 = 𝑐))
66	65	biimpd 229	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → ((𝑎‘0) = (𝑏‘0) → 𝑑 = 𝑐))
67	66	rexlimdvva 3211	. . . . . 6 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → (∃𝑎 ∈ (◡𝑆 “ {𝑑})∃𝑏 ∈ (◡𝑆 “ {𝑐})(𝑎‘0) = (𝑏‘0) → 𝑑 = 𝑐))
68	27, 67	syl5 34	. . . . 5 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → (𝑑 ∼ 𝑐 → 𝑑 = 𝑐))
69	26, 68	mpd 15	. . . 4 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → 𝑑 = 𝑐)
70	69	ex 412	. . 3 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) → ((𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴) → 𝑑 = 𝑐))
71	70	ralrimivva 3200	. 2 ⊢ (𝐴 ∈ 𝑊 → ∀𝑑 ∈ 𝐷 ∀𝑐 ∈ 𝐷 ((𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴) → 𝑑 = 𝑐))
72		breq1 5151	. . 3 ⊢ (𝑑 = 𝑐 → (𝑑 ∼ 𝐴 ↔ 𝑐 ∼ 𝐴))
73	72	reu4 3740	. 2 ⊢ (∃!𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ↔ (∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ∧ ∀𝑑 ∈ 𝐷 ∀𝑐 ∈ 𝐷 ((𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴) → 𝑑 = 𝑐)))
74	21, 71, 73	sylanbrc 583	1 ⊢ (𝐴 ∈ 𝑊 → ∃!𝑑 ∈ 𝐷 𝑑 ∼ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  ∃!wreu 3376  {crab 3433   ∖ cdif 3960  ∅c0 4339  {csn 4631  ⟨cop 4637  ⟨cotp 4639  ∪ ciun 4996   class class class wbr 5148   ↦ cmpt 5231   I cid 5582   × cxp 5687  ^◡ccnv 5688  dom cdm 5689  ran crn 5690   “ cima 5692   Fn wfn 6558  –onto→wfo 6561  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  1_oc1o 8498  2_oc2o 8499   Er wer 8741  0cc0 11153  1c1 11154   − cmin 11490  ...cfz 13544  ..^cfzo 13691  ♯chash 14366  Word cword 14549   splice csplice 14784  ⟨“cs2 14877   ~_FG cefg 19739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-ec 8746  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-concat 14606  df-s1 14631  df-substr 14676  df-pfx 14706  df-splice 14785  df-s2 14884  df-efg 19742

This theorem is referenced by:  efgred2  19786  frgpnabllem2  19907