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Theorem efgsfo 19757
Description: For any word, there is a sequence of extensions starting at a reduced word and ending at the target word, such that each word in the chain is an extension of the previous (inserting an element and its inverse at adjacent indices somewhere in the sequence). (Contributed by Mario Carneiro, 27-Sep-2015.)
Hypotheses
Ref Expression
efgval.w 𝑊 = ( I ‘Word (𝐼 × 2o))
efgval.r = ( ~FG𝐼)
efgval2.m 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
efgval2.t 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
efgred.d 𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))
efgred.s 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
Assertion
Ref Expression
efgsfo 𝑆:dom 𝑆onto𝑊
Distinct variable groups:   𝑦,𝑧   𝑡,𝑛,𝑣,𝑤,𝑦,𝑧,𝑚,𝑥   𝑚,𝑀   𝑥,𝑛,𝑀,𝑡,𝑣,𝑤   𝑘,𝑚,𝑡,𝑥,𝑇   𝑘,𝑛,𝑣,𝑤,𝑦,𝑧,𝑊,𝑚,𝑡,𝑥   ,𝑚,𝑡,𝑥,𝑦,𝑧   𝑚,𝐼,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧   𝐷,𝑚,𝑡
Allowed substitution hints:   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)   𝐼(𝑘)   𝑀(𝑦,𝑧,𝑘)

Proof of Theorem efgsfo
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑖 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . 4 𝑊 = ( I ‘Word (𝐼 × 2o))
2 efgval.r . . . 4 = ( ~FG𝐼)
3 efgval2.m . . . 4 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
4 efgval2.t . . . 4 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
5 efgred.d . . . 4 𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))
6 efgred.s . . . 4 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
71, 2, 3, 4, 5, 6efgsf 19747 . . 3 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊
87fdmi 6747 . . . 4 dom 𝑆 = {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}
98feq2i 6728 . . 3 (𝑆:dom 𝑆𝑊𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊)
107, 9mpbir 231 . 2 𝑆:dom 𝑆𝑊
11 frn 6743 . . . 4 (𝑆:dom 𝑆𝑊 → ran 𝑆𝑊)
1210, 11ax-mp 5 . . 3 ran 𝑆𝑊
13 fviss 6986 . . . . . . . . 9 ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
141, 13eqsstri 4030 . . . . . . . 8 𝑊 ⊆ Word (𝐼 × 2o)
1514sseli 3979 . . . . . . 7 (𝑐𝑊𝑐 ∈ Word (𝐼 × 2o))
16 lencl 14571 . . . . . . 7 (𝑐 ∈ Word (𝐼 × 2o) → (♯‘𝑐) ∈ ℕ0)
1715, 16syl 17 . . . . . 6 (𝑐𝑊 → (♯‘𝑐) ∈ ℕ0)
18 peano2nn0 12566 . . . . . 6 ((♯‘𝑐) ∈ ℕ0 → ((♯‘𝑐) + 1) ∈ ℕ0)
1914sseli 3979 . . . . . . . . . . . 12 (𝑎𝑊𝑎 ∈ Word (𝐼 × 2o))
20 lencl 14571 . . . . . . . . . . . 12 (𝑎 ∈ Word (𝐼 × 2o) → (♯‘𝑎) ∈ ℕ0)
2119, 20syl 17 . . . . . . . . . . 11 (𝑎𝑊 → (♯‘𝑎) ∈ ℕ0)
22 nn0nlt0 12552 . . . . . . . . . . . 12 ((♯‘𝑎) ∈ ℕ0 → ¬ (♯‘𝑎) < 0)
23 breq2 5147 . . . . . . . . . . . . 13 (𝑏 = 0 → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < 0))
2423notbid 318 . . . . . . . . . . . 12 (𝑏 = 0 → (¬ (♯‘𝑎) < 𝑏 ↔ ¬ (♯‘𝑎) < 0))
2522, 24imbitrrid 246 . . . . . . . . . . 11 (𝑏 = 0 → ((♯‘𝑎) ∈ ℕ0 → ¬ (♯‘𝑎) < 𝑏))
2621, 25syl5 34 . . . . . . . . . 10 (𝑏 = 0 → (𝑎𝑊 → ¬ (♯‘𝑎) < 𝑏))
2726ralrimiv 3145 . . . . . . . . 9 (𝑏 = 0 → ∀𝑎𝑊 ¬ (♯‘𝑎) < 𝑏)
28 rabeq0 4388 . . . . . . . . 9 ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} = ∅ ↔ ∀𝑎𝑊 ¬ (♯‘𝑎) < 𝑏)
2927, 28sylibr 234 . . . . . . . 8 (𝑏 = 0 → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} = ∅)
3029sseq1d 4015 . . . . . . 7 (𝑏 = 0 → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ ∅ ⊆ ran 𝑆))
31 breq2 5147 . . . . . . . . 9 (𝑏 = 𝑑 → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < 𝑑))
3231rabbidv 3444 . . . . . . . 8 (𝑏 = 𝑑 → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑})
3332sseq1d 4015 . . . . . . 7 (𝑏 = 𝑑 → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆))
34 breq2 5147 . . . . . . . . 9 (𝑏 = (𝑑 + 1) → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < (𝑑 + 1)))
3534rabbidv 3444 . . . . . . . 8 (𝑏 = (𝑑 + 1) → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)})
3635sseq1d 4015 . . . . . . 7 (𝑏 = (𝑑 + 1) → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆))
37 breq2 5147 . . . . . . . . 9 (𝑏 = ((♯‘𝑐) + 1) → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < ((♯‘𝑐) + 1)))
3837rabbidv 3444 . . . . . . . 8 (𝑏 = ((♯‘𝑐) + 1) → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)})
3938sseq1d 4015 . . . . . . 7 (𝑏 = ((♯‘𝑐) + 1) → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)} ⊆ ran 𝑆))
40 0ss 4400 . . . . . . 7 ∅ ⊆ ran 𝑆
41 simpr 484 . . . . . . . . . 10 ((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆)
42 fveqeq2 6915 . . . . . . . . . . . 12 (𝑎 = 𝑐 → ((♯‘𝑎) = 𝑑 ↔ (♯‘𝑐) = 𝑑))
4342cbvrabv 3447 . . . . . . . . . . 11 {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑} = {𝑐𝑊 ∣ (♯‘𝑐) = 𝑑}
44 eliun 4995 . . . . . . . . . . . . . . 15 (𝑐 𝑥𝑊 ran (𝑇𝑥) ↔ ∃𝑥𝑊 𝑐 ∈ ran (𝑇𝑥))
45 fveq2 6906 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑏 → (𝑇𝑥) = (𝑇𝑏))
4645rneqd 5949 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑏 → ran (𝑇𝑥) = ran (𝑇𝑏))
4746eleq2d 2827 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑏 → (𝑐 ∈ ran (𝑇𝑥) ↔ 𝑐 ∈ ran (𝑇𝑏)))
4847cbvrexvw 3238 . . . . . . . . . . . . . . 15 (∃𝑥𝑊 𝑐 ∈ ran (𝑇𝑥) ↔ ∃𝑏𝑊 𝑐 ∈ ran (𝑇𝑏))
4944, 48bitri 275 . . . . . . . . . . . . . 14 (𝑐 𝑥𝑊 ran (𝑇𝑥) ↔ ∃𝑏𝑊 𝑐 ∈ ran (𝑇𝑏))
50 simpl1r 1226 . . . . . . . . . . . . . . . . . 18 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆)
51 fveq2 6906 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑏 → (♯‘𝑎) = (♯‘𝑏))
5251breq1d 5153 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑏 → ((♯‘𝑎) < 𝑑 ↔ (♯‘𝑏) < 𝑑))
53 simprl 771 . . . . . . . . . . . . . . . . . . 19 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → 𝑏𝑊)
5414, 53sselid 3981 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → 𝑏 ∈ Word (𝐼 × 2o))
55 lencl 14571 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 ∈ Word (𝐼 × 2o) → (♯‘𝑏) ∈ ℕ0)
5654, 55syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → (♯‘𝑏) ∈ ℕ0)
5756nn0red 12588 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → (♯‘𝑏) ∈ ℝ)
58 2rp 13039 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℝ+
59 ltaddrp 13072 . . . . . . . . . . . . . . . . . . . . 21 (((♯‘𝑏) ∈ ℝ ∧ 2 ∈ ℝ+) → (♯‘𝑏) < ((♯‘𝑏) + 2))
6057, 58, 59sylancl 586 . . . . . . . . . . . . . . . . . . . 20 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → (♯‘𝑏) < ((♯‘𝑏) + 2))
611, 2, 3, 4efgtlen 19744 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) → (♯‘𝑐) = ((♯‘𝑏) + 2))
6261adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → (♯‘𝑐) = ((♯‘𝑏) + 2))
63 simpl3 1194 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → (♯‘𝑐) = 𝑑)
6462, 63eqtr3d 2779 . . . . . . . . . . . . . . . . . . . 20 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → ((♯‘𝑏) + 2) = 𝑑)
6560, 64breqtrd 5169 . . . . . . . . . . . . . . . . . . 19 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → (♯‘𝑏) < 𝑑)
6652, 53, 65elrabd 3694 . . . . . . . . . . . . . . . . . 18 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → 𝑏 ∈ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑})
6750, 66sseldd 3984 . . . . . . . . . . . . . . . . 17 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → 𝑏 ∈ ran 𝑆)
68 ffn 6736 . . . . . . . . . . . . . . . . . . 19 (𝑆:dom 𝑆𝑊𝑆 Fn dom 𝑆)
6910, 68ax-mp 5 . . . . . . . . . . . . . . . . . 18 𝑆 Fn dom 𝑆
70 fvelrnb 6969 . . . . . . . . . . . . . . . . . 18 (𝑆 Fn dom 𝑆 → (𝑏 ∈ ran 𝑆 ↔ ∃𝑜 ∈ dom 𝑆(𝑆𝑜) = 𝑏))
7169, 70ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ ran 𝑆 ↔ ∃𝑜 ∈ dom 𝑆(𝑆𝑜) = 𝑏)
7267, 71sylib 218 . . . . . . . . . . . . . . . 16 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → ∃𝑜 ∈ dom 𝑆(𝑆𝑜) = 𝑏)
73 simprrl 781 . . . . . . . . . . . . . . . . . . . 20 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → 𝑜 ∈ dom 𝑆)
741, 2, 3, 4, 5, 6efgsdm 19748 . . . . . . . . . . . . . . . . . . . . 21 (𝑜 ∈ dom 𝑆 ↔ (𝑜 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑜‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑜))(𝑜𝑖) ∈ ran (𝑇‘(𝑜‘(𝑖 − 1)))))
7574simp1bi 1146 . . . . . . . . . . . . . . . . . . . 20 (𝑜 ∈ dom 𝑆𝑜 ∈ (Word 𝑊 ∖ {∅}))
76 eldifi 4131 . . . . . . . . . . . . . . . . . . . 20 (𝑜 ∈ (Word 𝑊 ∖ {∅}) → 𝑜 ∈ Word 𝑊)
7773, 75, 763syl 18 . . . . . . . . . . . . . . . . . . 19 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → 𝑜 ∈ Word 𝑊)
78 simpl2 1193 . . . . . . . . . . . . . . . . . . 19 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → 𝑐𝑊)
79 simprlr 780 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → 𝑐 ∈ ran (𝑇𝑏))
80 simprrr 782 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → (𝑆𝑜) = 𝑏)
8180fveq2d 6910 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → (𝑇‘(𝑆𝑜)) = (𝑇𝑏))
8281rneqd 5949 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → ran (𝑇‘(𝑆𝑜)) = ran (𝑇𝑏))
8379, 82eleqtrrd 2844 . . . . . . . . . . . . . . . . . . . 20 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → 𝑐 ∈ ran (𝑇‘(𝑆𝑜)))
841, 2, 3, 4, 5, 6efgsp1 19755 . . . . . . . . . . . . . . . . . . . 20 ((𝑜 ∈ dom 𝑆𝑐 ∈ ran (𝑇‘(𝑆𝑜))) → (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆)
8573, 83, 84syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆)
861, 2, 3, 4, 5, 6efgsval2 19751 . . . . . . . . . . . . . . . . . . 19 ((𝑜 ∈ Word 𝑊𝑐𝑊 ∧ (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) = 𝑐)
8777, 78, 85, 86syl3anc 1373 . . . . . . . . . . . . . . . . . 18 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) = 𝑐)
88 fnfvelrn 7100 . . . . . . . . . . . . . . . . . . 19 ((𝑆 Fn dom 𝑆 ∧ (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) ∈ ran 𝑆)
8969, 85, 88sylancr 587 . . . . . . . . . . . . . . . . . 18 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) ∈ ran 𝑆)
9087, 89eqeltrrd 2842 . . . . . . . . . . . . . . . . 17 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → 𝑐 ∈ ran 𝑆)
9190anassrs 467 . . . . . . . . . . . . . . . 16 (((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏)) → 𝑐 ∈ ran 𝑆)
9272, 91rexlimddv 3161 . . . . . . . . . . . . . . 15 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → 𝑐 ∈ ran 𝑆)
9392rexlimdvaa 3156 . . . . . . . . . . . . . 14 (((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) → (∃𝑏𝑊 𝑐 ∈ ran (𝑇𝑏) → 𝑐 ∈ ran 𝑆))
9449, 93biimtrid 242 . . . . . . . . . . . . 13 (((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) → (𝑐 𝑥𝑊 ran (𝑇𝑥) → 𝑐 ∈ ran 𝑆))
95 eldif 3961 . . . . . . . . . . . . . . . . . . 19 (𝑐 ∈ (𝑊 𝑥𝑊 ran (𝑇𝑥)) ↔ (𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)))
965eleq2i 2833 . . . . . . . . . . . . . . . . . . . 20 (𝑐𝐷𝑐 ∈ (𝑊 𝑥𝑊 ran (𝑇𝑥)))
971, 2, 3, 4, 5, 6efgs1 19753 . . . . . . . . . . . . . . . . . . . 20 (𝑐𝐷 → ⟨“𝑐”⟩ ∈ dom 𝑆)
9896, 97sylbir 235 . . . . . . . . . . . . . . . . . . 19 (𝑐 ∈ (𝑊 𝑥𝑊 ran (𝑇𝑥)) → ⟨“𝑐”⟩ ∈ dom 𝑆)
9995, 98sylbir 235 . . . . . . . . . . . . . . . . . 18 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → ⟨“𝑐”⟩ ∈ dom 𝑆)
1001, 2, 3, 4, 5, 6efgsval 19749 . . . . . . . . . . . . . . . . . 18 (⟨“𝑐”⟩ ∈ dom 𝑆 → (𝑆‘⟨“𝑐”⟩) = (⟨“𝑐”⟩‘((♯‘⟨“𝑐”⟩) − 1)))
10199, 100syl 17 . . . . . . . . . . . . . . . . 17 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → (𝑆‘⟨“𝑐”⟩) = (⟨“𝑐”⟩‘((♯‘⟨“𝑐”⟩) − 1)))
102 s1len 14644 . . . . . . . . . . . . . . . . . . . . 21 (♯‘⟨“𝑐”⟩) = 1
103102oveq1i 7441 . . . . . . . . . . . . . . . . . . . 20 ((♯‘⟨“𝑐”⟩) − 1) = (1 − 1)
104 1m1e0 12338 . . . . . . . . . . . . . . . . . . . 20 (1 − 1) = 0
105103, 104eqtri 2765 . . . . . . . . . . . . . . . . . . 19 ((♯‘⟨“𝑐”⟩) − 1) = 0
106105fveq2i 6909 . . . . . . . . . . . . . . . . . 18 (⟨“𝑐”⟩‘((♯‘⟨“𝑐”⟩) − 1)) = (⟨“𝑐”⟩‘0)
107106a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → (⟨“𝑐”⟩‘((♯‘⟨“𝑐”⟩) − 1)) = (⟨“𝑐”⟩‘0))
108 s1fv 14648 . . . . . . . . . . . . . . . . . 18 (𝑐𝑊 → (⟨“𝑐”⟩‘0) = 𝑐)
109108adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → (⟨“𝑐”⟩‘0) = 𝑐)
110101, 107, 1093eqtrd 2781 . . . . . . . . . . . . . . . 16 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → (𝑆‘⟨“𝑐”⟩) = 𝑐)
111 fnfvelrn 7100 . . . . . . . . . . . . . . . . 17 ((𝑆 Fn dom 𝑆 ∧ ⟨“𝑐”⟩ ∈ dom 𝑆) → (𝑆‘⟨“𝑐”⟩) ∈ ran 𝑆)
11269, 99, 111sylancr 587 . . . . . . . . . . . . . . . 16 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → (𝑆‘⟨“𝑐”⟩) ∈ ran 𝑆)
113110, 112eqeltrrd 2842 . . . . . . . . . . . . . . 15 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → 𝑐 ∈ ran 𝑆)
114113ex 412 . . . . . . . . . . . . . 14 (𝑐𝑊 → (¬ 𝑐 𝑥𝑊 ran (𝑇𝑥) → 𝑐 ∈ ran 𝑆))
1151143ad2ant2 1135 . . . . . . . . . . . . 13 (((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) → (¬ 𝑐 𝑥𝑊 ran (𝑇𝑥) → 𝑐 ∈ ran 𝑆))
11694, 115pm2.61d 179 . . . . . . . . . . . 12 (((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) → 𝑐 ∈ ran 𝑆)
117116rabssdv 4075 . . . . . . . . . . 11 ((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑐𝑊 ∣ (♯‘𝑐) = 𝑑} ⊆ ran 𝑆)
11843, 117eqsstrid 4022 . . . . . . . . . 10 ((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑} ⊆ ran 𝑆)
11941, 118unssd 4192 . . . . . . . . 9 ((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑}) ⊆ ran 𝑆)
120119ex 412 . . . . . . . 8 (𝑑 ∈ ℕ0 → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆 → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑}) ⊆ ran 𝑆))
121 id 22 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ0𝑑 ∈ ℕ0)
122 nn0leltp1 12677 . . . . . . . . . . . . 13 (((♯‘𝑎) ∈ ℕ0𝑑 ∈ ℕ0) → ((♯‘𝑎) ≤ 𝑑 ↔ (♯‘𝑎) < (𝑑 + 1)))
12321, 121, 122syl2anr 597 . . . . . . . . . . . 12 ((𝑑 ∈ ℕ0𝑎𝑊) → ((♯‘𝑎) ≤ 𝑑 ↔ (♯‘𝑎) < (𝑑 + 1)))
12421nn0red 12588 . . . . . . . . . . . . 13 (𝑎𝑊 → (♯‘𝑎) ∈ ℝ)
125 nn0re 12535 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ0𝑑 ∈ ℝ)
126 leloe 11347 . . . . . . . . . . . . 13 (((♯‘𝑎) ∈ ℝ ∧ 𝑑 ∈ ℝ) → ((♯‘𝑎) ≤ 𝑑 ↔ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)))
127124, 125, 126syl2anr 597 . . . . . . . . . . . 12 ((𝑑 ∈ ℕ0𝑎𝑊) → ((♯‘𝑎) ≤ 𝑑 ↔ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)))
128123, 127bitr3d 281 . . . . . . . . . . 11 ((𝑑 ∈ ℕ0𝑎𝑊) → ((♯‘𝑎) < (𝑑 + 1) ↔ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)))
129128rabbidva 3443 . . . . . . . . . 10 (𝑑 ∈ ℕ0 → {𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} = {𝑎𝑊 ∣ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)})
130 unrab 4315 . . . . . . . . . 10 ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑}) = {𝑎𝑊 ∣ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)}
131129, 130eqtr4di 2795 . . . . . . . . 9 (𝑑 ∈ ℕ0 → {𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} = ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑}))
132131sseq1d 4015 . . . . . . . 8 (𝑑 ∈ ℕ0 → ({𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆 ↔ ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑}) ⊆ ran 𝑆))
133120, 132sylibrd 259 . . . . . . 7 (𝑑 ∈ ℕ0 → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆 → {𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆))
13430, 33, 36, 39, 40, 133nn0ind 12713 . . . . . 6 (((♯‘𝑐) + 1) ∈ ℕ0 → {𝑎𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)} ⊆ ran 𝑆)
13517, 18, 1343syl 18 . . . . 5 (𝑐𝑊 → {𝑎𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)} ⊆ ran 𝑆)
136 fveq2 6906 . . . . . . 7 (𝑎 = 𝑐 → (♯‘𝑎) = (♯‘𝑐))
137136breq1d 5153 . . . . . 6 (𝑎 = 𝑐 → ((♯‘𝑎) < ((♯‘𝑐) + 1) ↔ (♯‘𝑐) < ((♯‘𝑐) + 1)))
138 id 22 . . . . . 6 (𝑐𝑊𝑐𝑊)
13917nn0red 12588 . . . . . . 7 (𝑐𝑊 → (♯‘𝑐) ∈ ℝ)
140139ltp1d 12198 . . . . . 6 (𝑐𝑊 → (♯‘𝑐) < ((♯‘𝑐) + 1))
141137, 138, 140elrabd 3694 . . . . 5 (𝑐𝑊𝑐 ∈ {𝑎𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)})
142135, 141sseldd 3984 . . . 4 (𝑐𝑊𝑐 ∈ ran 𝑆)
143142ssriv 3987 . . 3 𝑊 ⊆ ran 𝑆
14412, 143eqssi 4000 . 2 ran 𝑆 = 𝑊
145 dffo2 6824 . 2 (𝑆:dom 𝑆onto𝑊 ↔ (𝑆:dom 𝑆𝑊 ∧ ran 𝑆 = 𝑊))
14610, 144, 145mpbir2an 711 1 𝑆:dom 𝑆onto𝑊
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  cdif 3948  cun 3949  wss 3951  c0 4333  {csn 4626  cop 4632  cotp 4634   ciun 4991   class class class wbr 5143  cmpt 5225   I cid 5577   × cxp 5683  dom cdm 5685  ran crn 5686   Fn wfn 6556  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431  cmpo 7433  1oc1o 8499  2oc2o 8500  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   < clt 11295  cle 11296  cmin 11492  2c2 12321  0cn0 12526  +crp 13034  ...cfz 13547  ..^cfzo 13694  chash 14369  Word cword 14552   ++ cconcat 14608  ⟨“cs1 14633   splice csplice 14787  ⟨“cs2 14880   ~FG cefg 19724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-substr 14679  df-pfx 14709  df-splice 14788  df-s2 14887
This theorem is referenced by:  efgredlemc  19763  efgrelexlemb  19768  efgredeu  19770  efgred2  19771
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