Metamath Proof Explorer

Theorem 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 × 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 efgredeu ( 𝐴 ∈ 𝑊 → ∃! 𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 )

Proof

Step Hyp Ref Expression
1 efgval.w ⊢ 𝑊 = ( I ‘ Word ( 𝐼 × 2o ) )
2 efgval.r ⊢ ∼ = ( ~FG ‘ 𝐼 )
3 efgval2.m ⊢ 𝑀 = ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ ⟨ 𝑦 , ( 1o ∖ 𝑧 ) ⟩ )
4 efgval2.t ⊢ 𝑇 = ( 𝑣 ∈ 𝑊 ↦ ( 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) , 𝑤 ∈ ( 𝐼 × 2o ) ↦ ( 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 )