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Metamath Proof Explorer

Theorem efgsdm

Description: Elementhood in the domain of S , the set of sequences of extensions starting at an irreducible word. (Contributed by Mario Carneiro, 27-Sep-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 efgsdm ( 𝐹 ∈ dom 𝑆 ↔ ( 𝐹 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) ) )

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 fveq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 0 ) = ( 𝐹 ‘ 0 ) )
8 7 eleq1d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 0 ) ∈ 𝐷 ↔ ( 𝐹 ‘ 0 ) ∈ 𝐷 ) )
9 fveq2 ⊢ ( 𝑓 = 𝐹 → ( ♯ ‘ 𝑓 ) = ( ♯ ‘ 𝐹 ) )
10 9 oveq2d ⊢ ( 𝑓 = 𝐹 → ( 1 ..^ ( ♯ ‘ 𝑓 ) ) = ( 1 ..^ ( ♯ ‘ 𝐹 ) ) )
11 fveq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑖 ) = ( 𝐹 ‘ 𝑖 ) )
12 fveq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ ( 𝑖 − 1 ) ) = ( 𝐹 ‘ ( 𝑖 − 1 ) ) )
13 12 fveq2d ⊢ ( 𝑓 = 𝐹 → ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) = ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) )
14 13 rneqd ⊢ ( 𝑓 = 𝐹 → ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) = ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) )
15 11 14 eleq12d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ↔ ( 𝐹 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) ) )
16 10 15 raleqbidv ⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ↔ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) ) )
17 8 16 anbi12d ⊢ ( 𝑓 = 𝐹 → ( ( ( 𝑓 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ) ↔ ( ( 𝐹 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) ) ) )
18 1 2 3 4 5 6 efgsf ⊢ 𝑆 : { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } ⟶ 𝑊
19 18 fdmi ⊢ dom 𝑆 = { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) }
20 fveq1 ⊢ ( 𝑡 = 𝑓 → ( 𝑡 ‘ 0 ) = ( 𝑓 ‘ 0 ) )
21 20 eleq1d ⊢ ( 𝑡 = 𝑓 → ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ↔ ( 𝑓 ‘ 0 ) ∈ 𝐷 ) )
22 fveq2 ⊢ ( 𝑘 = 𝑖 → ( 𝑡 ‘ 𝑘 ) = ( 𝑡 ‘ 𝑖 ) )
23 fvoveq1 ⊢ ( 𝑘 = 𝑖 → ( 𝑡 ‘ ( 𝑘 − 1 ) ) = ( 𝑡 ‘ ( 𝑖 − 1 ) ) )
24 23 fveq2d ⊢ ( 𝑘 = 𝑖 → ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) = ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑖 − 1 ) ) ) )
25 24 rneqd ⊢ ( 𝑘 = 𝑖 → ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) = ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑖 − 1 ) ) ) )
26 22 25 eleq12d ⊢ ( 𝑘 = 𝑖 → ( ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ↔ ( 𝑡 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑖 − 1 ) ) ) ) )
27 26 cbvralvw ⊢ ( ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ↔ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑖 − 1 ) ) ) )
28 fveq2 ⊢ ( 𝑡 = 𝑓 → ( ♯ ‘ 𝑡 ) = ( ♯ ‘ 𝑓 ) )
29 28 oveq2d ⊢ ( 𝑡 = 𝑓 → ( 1 ..^ ( ♯ ‘ 𝑡 ) ) = ( 1 ..^ ( ♯ ‘ 𝑓 ) ) )
30 fveq1 ⊢ ( 𝑡 = 𝑓 → ( 𝑡 ‘ 𝑖 ) = ( 𝑓 ‘ 𝑖 ) )
31 fveq1 ⊢ ( 𝑡 = 𝑓 → ( 𝑡 ‘ ( 𝑖 − 1 ) ) = ( 𝑓 ‘ ( 𝑖 − 1 ) ) )
32 31 fveq2d ⊢ ( 𝑡 = 𝑓 → ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑖 − 1 ) ) ) = ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) )
33 32 rneqd ⊢ ( 𝑡 = 𝑓 → ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑖 − 1 ) ) ) = ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) )
34 30 33 eleq12d ⊢ ( 𝑡 = 𝑓 → ( ( 𝑡 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑖 − 1 ) ) ) ↔ ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ) )
35 29 34 raleqbidv ⊢ ( 𝑡 = 𝑓 → ( ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑖 − 1 ) ) ) ↔ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ) )
36 27 35 bitrid ⊢ ( 𝑡 = 𝑓 → ( ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ↔ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ) )
37 21 36 anbi12d ⊢ ( 𝑡 = 𝑓 → ( ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) ↔ ( ( 𝑓 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ) ) )
38 37 cbvrabv ⊢ { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } = { 𝑓 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑓 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ) }
39 19 38 eqtri ⊢ dom 𝑆 = { 𝑓 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑓 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ) }
40 17 39 elrab2 ⊢ ( 𝐹 ∈ dom 𝑆 ↔ ( 𝐹 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( ( 𝐹 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) ) ) )
41 3anass ⊢ ( ( 𝐹 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) ) ↔ ( 𝐹 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( ( 𝐹 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) ) ) )
42 40 41 bitr4i ⊢ ( 𝐹 ∈ dom 𝑆 ↔ ( 𝐹 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) ) )