Theorem efgredeu 19727

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 19714	. . . 4 ⊢ 𝑆:dom 𝑆–onto→𝑊
8		foelrn 7059	. . . 4 ⊢ ((𝑆:dom 𝑆–onto→𝑊 ∧ 𝐴 ∈ 𝑊) → ∃𝑎 ∈ dom 𝑆 𝐴 = (𝑆‘𝑎))
9	7, 8	mpan 691	. . 3 ⊢ (𝐴 ∈ 𝑊 → ∃𝑎 ∈ dom 𝑆 𝐴 = (𝑆‘𝑎))
10	1, 2, 3, 4, 5, 6	efgsdm 19705	. . . . . . 7 ⊢ (𝑎 ∈ dom 𝑆 ↔ (𝑎 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑎‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑎))(𝑎‘𝑖) ∈ ran (𝑇‘(𝑎‘(𝑖 − 1)))))
11	10	simp2bi 1147	. . . . . 6 ⊢ (𝑎 ∈ dom 𝑆 → (𝑎‘0) ∈ 𝐷)
12	1, 2, 3, 4, 5, 6	efgsrel 19709	. . . . . . 7 ⊢ (𝑎 ∈ dom 𝑆 → (𝑎‘0) ∼ (𝑆‘𝑎))
13	12	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆) → (𝑎‘0) ∼ (𝑆‘𝑎))
14		breq1 5088	. . . . . . 7 ⊢ (𝑑 = (𝑎‘0) → (𝑑 ∼ (𝑆‘𝑎) ↔ (𝑎‘0) ∼ (𝑆‘𝑎)))
15	14	rspcev 3564	. . . . . 6 ⊢ (((𝑎‘0) ∈ 𝐷 ∧ (𝑎‘0) ∼ (𝑆‘𝑎)) → ∃𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝑎))
16	11, 13, 15	syl2an2 687	. . . . 5 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆) → ∃𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝑎))
17		breq2 5089	. . . . . 6 ⊢ (𝐴 = (𝑆‘𝑎) → (𝑑 ∼ 𝐴 ↔ 𝑑 ∼ (𝑆‘𝑎)))
18	17	rexbidv 3161	. . . . 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 3138	. . 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 19693	. . . . . . 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 8663	. . . . 5 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → 𝑑 ∼ 𝑐)
27	1, 2, 3, 4, 5, 6	efgrelex 19726	. . . . . 6 ⊢ (𝑑 ∼ 𝑐 → ∃𝑎 ∈ (◡𝑆 “ {𝑑})∃𝑏 ∈ (◡𝑆 “ {𝑐})(𝑎‘0) = (𝑏‘0))
28		fofn 6754	. . . . . . . . . . . . . 14 ⊢ (𝑆:dom 𝑆–onto→𝑊 → 𝑆 Fn dom 𝑆)
29		fniniseg 7012	. . . . . . . . . . . . . 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 496	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (◡𝑆 “ {𝑑}) → 𝑎 ∈ dom 𝑆)
32	31	ad2antrl 729	. . . . . . . . . . 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 19706	. . . . . . . . . . 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 497	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (◡𝑆 “ {𝑑}) → (𝑆‘𝑎) = 𝑑)
36	35	ad2antrl 729	. . . . . . . . . 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 2836	. . . . . . . . . . . . . 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 19711	. . . . . . . . . . . . . . 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 7382	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → ((♯‘𝑎) − 1) = (1 − 1))
44		1m1e0 12253	. . . . . . . . . . . 12 ⊢ (1 − 1) = 0
45	43, 44	eqtrdi 2787	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → ((♯‘𝑎) − 1) = 0)
46	45	fveq2d 6844	. . . . . . . . . 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 2780	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑎‘0) = 𝑑)
48		fniniseg 7012	. . . . . . . . . . . . . 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 496	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (◡𝑆 “ {𝑐}) → 𝑏 ∈ dom 𝑆)
51	50	ad2antll 730	. . . . . . . . . . 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 19706	. . . . . . . . . . 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 497	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (◡𝑆 “ {𝑐}) → (𝑆‘𝑏) = 𝑐)
55	54	ad2antll 730	. . . . . . . . . 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 2836	. . . . . . . . . . . . . 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 19711	. . . . . . . . . . . . . . 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 7382	. . . . . . . . . . . 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 2787	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → ((♯‘𝑏) − 1) = 0)
63	62	fveq2d 6844	. . . . . . . . . 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 2780	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑏‘0) = 𝑐)
65	47, 64	eqeq12d 2752	. . . . . . . 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 3194	. . . . . 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 3180	. 2 ⊢ (𝐴 ∈ 𝑊 → ∀𝑑 ∈ 𝐷 ∀𝑐 ∈ 𝐷 ((𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴) → 𝑑 = 𝑐))
72		breq1 5088	. . 3 ⊢ (𝑑 = 𝑐 → (𝑑 ∼ 𝐴 ↔ 𝑐 ∼ 𝐴))
73	72	reu4 3677	. 2 ⊢ (∃!𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ↔ (∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ∧ ∀𝑑 ∈ 𝐷 ∀𝑐 ∈ 𝐷 ((𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴) → 𝑑 = 𝑐)))
74	21, 71, 73	sylanbrc 584	1 ⊢ (𝐴 ∈ 𝑊 → ∃!𝑑 ∈ 𝐷 𝑑 ∼ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  ∃!wreu 3340  {crab 3389   ∖ cdif 3886  ∅c0 4273  {csn 4567  ⟨cop 4573  ⟨cotp 4575  ∪ ciun 4933   class class class wbr 5085   ↦ cmpt 5166   I cid 5525   × cxp 5629  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   “ cima 5634   Fn wfn 6493  –onto→wfo 6496  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  1_oc1o 8398  2_oc2o 8399   Er wer 8640  0cc0 11038  1c1 11039   − cmin 11377  ...cfz 13461  ..^cfzo 13608  ♯chash 14292  Word cword 14475   splice csplice 14711  ⟨“cs2 14803   ~_FG cefg 19681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-ec 8645  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-rp 12943  df-fz 13462  df-fzo 13609  df-hash 14293  df-word 14476  df-concat 14533  df-s1 14559  df-substr 14604  df-pfx 14634  df-splice 14712  df-s2 14810  df-efg 19684

This theorem is referenced by:  efgred2  19728  frgpnabllem2  19849