efgredeu

Description: There is a unique reduced word equivalent to a given word. (Contributed by Mario Carneiro, 1-Oct-2015)

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

Proof

Step	Hyp	Ref	Expression
1		efgval.w	⊢ 𝑊 = ( I ‘ Word ( 𝐼 × 2_o ) )
2		efgval.r	⊢ ∼ = ( ~_FG ‘ 𝐼 )
3		efgval2.m	⊢ 𝑀 = ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2_o ↦ ⟨ 𝑦 , ( 1_o ∖ 𝑧 ) ⟩ )
4		efgval2.t	⊢ 𝑇 = ( 𝑣 ∈ 𝑊 ↦ ( 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) , 𝑤 ∈ ( 𝐼 × 2_o ) ↦ ( 𝑣 splice ⟨ 𝑛 , 𝑛 , ⟨“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”⟩ ⟩ ) ) )
5		efgred.d	⊢ 𝐷 = ( 𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran ( 𝑇 ‘ 𝑥 ) )
6		efgred.s	⊢ 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	⊢ 𝑆 : dom 𝑆 –onto→ 𝑊
8		foelrn	⊢ ( ( 𝑆 : dom 𝑆 –onto→ 𝑊 ∧ 𝐴 ∈ 𝑊 ) → ∃ 𝑎 ∈ dom 𝑆 𝐴 = ( 𝑆 ‘ 𝑎 ) )
9	7 8	mpan	⊢ ( 𝐴 ∈ 𝑊 → ∃ 𝑎 ∈ dom 𝑆 𝐴 = ( 𝑆 ‘ 𝑎 ) )
10	1 2 3 4 5 6	efgsdm	⊢ ( 𝑎 ∈ dom 𝑆 ↔ ( 𝑎 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( 𝑎 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑎 ) ) ( 𝑎 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑎 ‘ ( 𝑖 − 1 ) ) ) ) )
11	10	simp2bi	⊢ ( 𝑎 ∈ dom 𝑆 → ( 𝑎 ‘ 0 ) ∈ 𝐷 )
12	1 2 3 4 5 6	efgsrel	⊢ ( 𝑎 ∈ dom 𝑆 → ( 𝑎 ‘ 0 ) ∼ ( 𝑆 ‘ 𝑎 ) )
13	12	adantl	⊢ ( ( 𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆 ) → ( 𝑎 ‘ 0 ) ∼ ( 𝑆 ‘ 𝑎 ) )
14		breq1	⊢ ( 𝑑 = ( 𝑎 ‘ 0 ) → ( 𝑑 ∼ ( 𝑆 ‘ 𝑎 ) ↔ ( 𝑎 ‘ 0 ) ∼ ( 𝑆 ‘ 𝑎 ) ) )
15	14	rspcev	⊢ ( ( ( 𝑎 ‘ 0 ) ∈ 𝐷 ∧ ( 𝑎 ‘ 0 ) ∼ ( 𝑆 ‘ 𝑎 ) ) → ∃ 𝑑 ∈ 𝐷 𝑑 ∼ ( 𝑆 ‘ 𝑎 ) )
16	11 13 15	syl2an2	⊢ ( ( 𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆 ) → ∃ 𝑑 ∈ 𝐷 𝑑 ∼ ( 𝑆 ‘ 𝑎 ) )
17		breq2	⊢ ( 𝐴 = ( 𝑆 ‘ 𝑎 ) → ( 𝑑 ∼ 𝐴 ↔ 𝑑 ∼ ( 𝑆 ‘ 𝑎 ) ) )
18	17	rexbidv	⊢ ( 𝐴 = ( 𝑆 ‘ 𝑎 ) → ( ∃ 𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ↔ ∃ 𝑑 ∈ 𝐷 𝑑 ∼ ( 𝑆 ‘ 𝑎 ) ) )
19	16 18	syl5ibrcom	⊢ ( ( 𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆 ) → ( 𝐴 = ( 𝑆 ‘ 𝑎 ) → ∃ 𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ) )
20	19	rexlimdva	⊢ ( 𝐴 ∈ 𝑊 → ( ∃ 𝑎 ∈ dom 𝑆 𝐴 = ( 𝑆 ‘ 𝑎 ) → ∃ 𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ) )
21	9 20	mpd	⊢ ( 𝐴 ∈ 𝑊 → ∃ 𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 )
22	1 2	efger	⊢ ∼ Er 𝑊
23	22	a1i	⊢ ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) → ∼ Er 𝑊 )
24		simprl	⊢ ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) → 𝑑 ∼ 𝐴 )
25		simprr	⊢ ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) → 𝑐 ∼ 𝐴 )
26	23 24 25	ertr4d	⊢ ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) → 𝑑 ∼ 𝑐 )
27	1 2 3 4 5 6	efgrelex	⊢ ( 𝑑 ∼ 𝑐 → ∃ 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ∃ 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) )
28		fofn	⊢ ( 𝑆 : dom 𝑆 –onto→ 𝑊 → 𝑆 Fn dom 𝑆 )
29		fniniseg	⊢ ( 𝑆 Fn dom 𝑆 → ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ↔ ( 𝑎 ∈ dom 𝑆 ∧ ( 𝑆 ‘ 𝑎 ) = 𝑑 ) ) )
30	7 28 29	mp2b	⊢ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ↔ ( 𝑎 ∈ dom 𝑆 ∧ ( 𝑆 ‘ 𝑎 ) = 𝑑 ) )
31	30	simplbi	⊢ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) → 𝑎 ∈ dom 𝑆 )
32	31	ad2antrl	⊢ ( ( ( ( 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	⊢ ( 𝑎 ∈ dom 𝑆 → ( 𝑆 ‘ 𝑎 ) = ( 𝑎 ‘ ( ( ♯ ‘ 𝑎 ) − 1 ) ) )
34	32 33	syl	⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑆 ‘ 𝑎 ) = ( 𝑎 ‘ ( ( ♯ ‘ 𝑎 ) − 1 ) ) )
35	30	simprbi	⊢ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) → ( 𝑆 ‘ 𝑎 ) = 𝑑 )
36	35	ad2antrl	⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑆 ‘ 𝑎 ) = 𝑑 )
37		simpllr	⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) )
38	37	simpld	⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ) ) → 𝑑 ∈ 𝐷 )
39	36 38	eqeltrd	⊢ ( ( ( ( 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	⊢ ( 𝑎 ∈ dom 𝑆 → ( ( 𝑆 ‘ 𝑎 ) ∈ 𝐷 ↔ ( ♯ ‘ 𝑎 ) = 1 ) )
41	32 40	syl	⊢ ( ( ( ( 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	⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ) ) → ( ♯ ‘ 𝑎 ) = 1 )
43	42	oveq1d	⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ) ) → ( ( ♯ ‘ 𝑎 ) − 1 ) = ( 1 − 1 ) )
44		1m1e0	⊢ ( 1 − 1 ) = 0
45	43 44	eqtrdi	⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ) ) → ( ( ♯ ‘ 𝑎 ) − 1 ) = 0 )
46	45	fveq2d	⊢ ( ( ( ( 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	⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑎 ‘ 0 ) = 𝑑 )
48		fniniseg	⊢ ( 𝑆 Fn dom 𝑆 → ( 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ↔ ( 𝑏 ∈ dom 𝑆 ∧ ( 𝑆 ‘ 𝑏 ) = 𝑐 ) ) )
49	7 28 48	mp2b	⊢ ( 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ↔ ( 𝑏 ∈ dom 𝑆 ∧ ( 𝑆 ‘ 𝑏 ) = 𝑐 ) )
50	49	simplbi	⊢ ( 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) → 𝑏 ∈ dom 𝑆 )
51	50	ad2antll	⊢ ( ( ( ( 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	⊢ ( 𝑏 ∈ dom 𝑆 → ( 𝑆 ‘ 𝑏 ) = ( 𝑏 ‘ ( ( ♯ ‘ 𝑏 ) − 1 ) ) )
53	51 52	syl	⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑆 ‘ 𝑏 ) = ( 𝑏 ‘ ( ( ♯ ‘ 𝑏 ) − 1 ) ) )
54	49	simprbi	⊢ ( 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) → ( 𝑆 ‘ 𝑏 ) = 𝑐 )
55	54	ad2antll	⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑆 ‘ 𝑏 ) = 𝑐 )
56	37	simprd	⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ) ) → 𝑐 ∈ 𝐷 )
57	55 56	eqeltrd	⊢ ( ( ( ( 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	⊢ ( 𝑏 ∈ dom 𝑆 → ( ( 𝑆 ‘ 𝑏 ) ∈ 𝐷 ↔ ( ♯ ‘ 𝑏 ) = 1 ) )
59	51 58	syl	⊢ ( ( ( ( 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	⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ) ) → ( ♯ ‘ 𝑏 ) = 1 )
61	60	oveq1d	⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ) ) → ( ( ♯ ‘ 𝑏 ) − 1 ) = ( 1 − 1 ) )
62	61 44	eqtrdi	⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ) ) → ( ( ♯ ‘ 𝑏 ) − 1 ) = 0 )
63	62	fveq2d	⊢ ( ( ( ( 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	⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ^◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ^◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑏 ‘ 0 ) = 𝑐 )
65	47 64	eqeq12d	⊢ ( ( ( ( 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	⊢ ( ( ( ( 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	⊢ ( ( ( 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	⊢ ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) → ( 𝑑 ∼ 𝑐 → 𝑑 = 𝑐 ) )
69	26 68	mpd	⊢ ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) → 𝑑 = 𝑐 )
70	69	ex	⊢ ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) → ( ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) → 𝑑 = 𝑐 ) )
71	70	ralrimivva	⊢ ( 𝐴 ∈ 𝑊 → ∀ 𝑑 ∈ 𝐷 ∀ 𝑐 ∈ 𝐷 ( ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) → 𝑑 = 𝑐 ) )
72		breq1	⊢ ( 𝑑 = 𝑐 → ( 𝑑 ∼ 𝐴 ↔ 𝑐 ∼ 𝐴 ) )
73	72	reu4	⊢ ( ∃! 𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ↔ ( ∃ 𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ∧ ∀ 𝑑 ∈ 𝐷 ∀ 𝑐 ∈ 𝐷 ( ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) → 𝑑 = 𝑐 ) ) )
74	21 71 73	sylanbrc	⊢ ( 𝐴 ∈ 𝑊 → ∃! 𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 )

Metamath Proof Explorer

Theorem efgredeu

Proof