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Theorem efgsfo 19521
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 19511 . . 3 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊
87fdmi 6680 . . . 4 dom 𝑆 = {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}
98feq2i 6660 . . 3 (𝑆:dom 𝑆𝑊𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊)
107, 9mpbir 230 . 2 𝑆:dom 𝑆𝑊
11 frn 6675 . . . 4 (𝑆:dom 𝑆𝑊 → ran 𝑆𝑊)
1210, 11ax-mp 5 . . 3 ran 𝑆𝑊
13 fviss 6918 . . . . . . . . 9 ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
141, 13eqsstri 3978 . . . . . . . 8 𝑊 ⊆ Word (𝐼 × 2o)
1514sseli 3940 . . . . . . 7 (𝑐𝑊𝑐 ∈ Word (𝐼 × 2o))
16 lencl 14421 . . . . . . 7 (𝑐 ∈ Word (𝐼 × 2o) → (♯‘𝑐) ∈ ℕ0)
1715, 16syl 17 . . . . . 6 (𝑐𝑊 → (♯‘𝑐) ∈ ℕ0)
18 peano2nn0 12453 . . . . . 6 ((♯‘𝑐) ∈ ℕ0 → ((♯‘𝑐) + 1) ∈ ℕ0)
1914sseli 3940 . . . . . . . . . . . 12 (𝑎𝑊𝑎 ∈ Word (𝐼 × 2o))
20 lencl 14421 . . . . . . . . . . . 12 (𝑎 ∈ Word (𝐼 × 2o) → (♯‘𝑎) ∈ ℕ0)
2119, 20syl 17 . . . . . . . . . . 11 (𝑎𝑊 → (♯‘𝑎) ∈ ℕ0)
22 nn0nlt0 12439 . . . . . . . . . . . 12 ((♯‘𝑎) ∈ ℕ0 → ¬ (♯‘𝑎) < 0)
23 breq2 5109 . . . . . . . . . . . . 13 (𝑏 = 0 → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < 0))
2423notbid 317 . . . . . . . . . . . 12 (𝑏 = 0 → (¬ (♯‘𝑎) < 𝑏 ↔ ¬ (♯‘𝑎) < 0))
2522, 24syl5ibr 245 . . . . . . . . . . 11 (𝑏 = 0 → ((♯‘𝑎) ∈ ℕ0 → ¬ (♯‘𝑎) < 𝑏))
2621, 25syl5 34 . . . . . . . . . 10 (𝑏 = 0 → (𝑎𝑊 → ¬ (♯‘𝑎) < 𝑏))
2726ralrimiv 3142 . . . . . . . . 9 (𝑏 = 0 → ∀𝑎𝑊 ¬ (♯‘𝑎) < 𝑏)
28 rabeq0 4344 . . . . . . . . 9 ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} = ∅ ↔ ∀𝑎𝑊 ¬ (♯‘𝑎) < 𝑏)
2927, 28sylibr 233 . . . . . . . 8 (𝑏 = 0 → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} = ∅)
3029sseq1d 3975 . . . . . . 7 (𝑏 = 0 → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ ∅ ⊆ ran 𝑆))
31 breq2 5109 . . . . . . . . 9 (𝑏 = 𝑑 → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < 𝑑))
3231rabbidv 3415 . . . . . . . 8 (𝑏 = 𝑑 → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑})
3332sseq1d 3975 . . . . . . 7 (𝑏 = 𝑑 → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆))
34 breq2 5109 . . . . . . . . 9 (𝑏 = (𝑑 + 1) → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < (𝑑 + 1)))
3534rabbidv 3415 . . . . . . . 8 (𝑏 = (𝑑 + 1) → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)})
3635sseq1d 3975 . . . . . . 7 (𝑏 = (𝑑 + 1) → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆))
37 breq2 5109 . . . . . . . . 9 (𝑏 = ((♯‘𝑐) + 1) → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < ((♯‘𝑐) + 1)))
3837rabbidv 3415 . . . . . . . 8 (𝑏 = ((♯‘𝑐) + 1) → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)})
3938sseq1d 3975 . . . . . . 7 (𝑏 = ((♯‘𝑐) + 1) → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)} ⊆ ran 𝑆))
40 0ss 4356 . . . . . . 7 ∅ ⊆ ran 𝑆
41 simpr 485 . . . . . . . . . 10 ((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆)
42 fveqeq2 6851 . . . . . . . . . . . 12 (𝑎 = 𝑐 → ((♯‘𝑎) = 𝑑 ↔ (♯‘𝑐) = 𝑑))
4342cbvrabv 3417 . . . . . . . . . . 11 {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑} = {𝑐𝑊 ∣ (♯‘𝑐) = 𝑑}
44 eliun 4958 . . . . . . . . . . . . . . 15 (𝑐 𝑥𝑊 ran (𝑇𝑥) ↔ ∃𝑥𝑊 𝑐 ∈ ran (𝑇𝑥))
45 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑏 → (𝑇𝑥) = (𝑇𝑏))
4645rneqd 5893 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑏 → ran (𝑇𝑥) = ran (𝑇𝑏))
4746eleq2d 2823 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑏 → (𝑐 ∈ ran (𝑇𝑥) ↔ 𝑐 ∈ ran (𝑇𝑏)))
4847cbvrexvw 3226 . . . . . . . . . . . . . . 15 (∃𝑥𝑊 𝑐 ∈ ran (𝑇𝑥) ↔ ∃𝑏𝑊 𝑐 ∈ ran (𝑇𝑏))
4944, 48bitri 274 . . . . . . . . . . . . . 14 (𝑐 𝑥𝑊 ran (𝑇𝑥) ↔ ∃𝑏𝑊 𝑐 ∈ ran (𝑇𝑏))
50 simpl1r 1225 . . . . . . . . . . . . . . . . . 18 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆)
51 fveq2 6842 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑏 → (♯‘𝑎) = (♯‘𝑏))
5251breq1d 5115 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑏 → ((♯‘𝑎) < 𝑑 ↔ (♯‘𝑏) < 𝑑))
53 simprl 769 . . . . . . . . . . . . . . . . . . 19 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → 𝑏𝑊)
5414, 53sselid 3942 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → 𝑏 ∈ Word (𝐼 × 2o))
55 lencl 14421 . . . . . . . . . . . . . . . . . . . . . . 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 12474 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → (♯‘𝑏) ∈ ℝ)
58 2rp 12920 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℝ+
59 ltaddrp 12952 . . . . . . . . . . . . . . . . . . . . 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 19508 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) → (♯‘𝑐) = ((♯‘𝑏) + 2))
6261adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → (♯‘𝑐) = ((♯‘𝑏) + 2))
63 simpl3 1193 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → (♯‘𝑐) = 𝑑)
6462, 63eqtr3d 2778 . . . . . . . . . . . . . . . . . . . 20 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → ((♯‘𝑏) + 2) = 𝑑)
6560, 64breqtrd 5131 . . . . . . . . . . . . . . . . . . 19 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → (♯‘𝑏) < 𝑑)
6652, 53, 65elrabd 3647 . . . . . . . . . . . . . . . . . 18 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → 𝑏 ∈ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑})
6750, 66sseldd 3945 . . . . . . . . . . . . . . . . 17 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → 𝑏 ∈ ran 𝑆)
68 ffn 6668 . . . . . . . . . . . . . . . . . . 19 (𝑆:dom 𝑆𝑊𝑆 Fn dom 𝑆)
6910, 68ax-mp 5 . . . . . . . . . . . . . . . . . 18 𝑆 Fn dom 𝑆
70 fvelrnb 6903 . . . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . . 16 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → ∃𝑜 ∈ dom 𝑆(𝑆𝑜) = 𝑏)
73 simprrl 779 . . . . . . . . . . . . . . . . . . . 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 19512 . . . . . . . . . . . . . . . . . . . . 21 (𝑜 ∈ dom 𝑆 ↔ (𝑜 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑜‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑜))(𝑜𝑖) ∈ ran (𝑇‘(𝑜‘(𝑖 − 1)))))
7574simp1bi 1145 . . . . . . . . . . . . . . . . . . . 20 (𝑜 ∈ dom 𝑆𝑜 ∈ (Word 𝑊 ∖ {∅}))
76 eldifi 4086 . . . . . . . . . . . . . . . . . . . 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 1192 . . . . . . . . . . . . . . . . . . 19 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → 𝑐𝑊)
79 simprlr 778 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → 𝑐 ∈ ran (𝑇𝑏))
80 simprrr 780 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → (𝑆𝑜) = 𝑏)
8180fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → (𝑇‘(𝑆𝑜)) = (𝑇𝑏))
8281rneqd 5893 . . . . . . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . . . . . . 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 19519 . . . . . . . . . . . . . . . . . . . 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 19515 . . . . . . . . . . . . . . . . . . 19 ((𝑜 ∈ Word 𝑊𝑐𝑊 ∧ (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) = 𝑐)
8777, 78, 85, 86syl3anc 1371 . . . . . . . . . . . . . . . . . 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 7031 . . . . . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . . . . 17 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → 𝑐 ∈ ran 𝑆)
9190anassrs 468 . . . . . . . . . . . . . . . 16 (((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏)) → 𝑐 ∈ ran 𝑆)
9272, 91rexlimddv 3158 . . . . . . . . . . . . . . 15 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → 𝑐 ∈ ran 𝑆)
9392rexlimdvaa 3153 . . . . . . . . . . . . . 14 (((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) → (∃𝑏𝑊 𝑐 ∈ ran (𝑇𝑏) → 𝑐 ∈ ran 𝑆))
9449, 93biimtrid 241 . . . . . . . . . . . . 13 (((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) → (𝑐 𝑥𝑊 ran (𝑇𝑥) → 𝑐 ∈ ran 𝑆))
95 eldif 3920 . . . . . . . . . . . . . . . . . . 19 (𝑐 ∈ (𝑊 𝑥𝑊 ran (𝑇𝑥)) ↔ (𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)))
965eleq2i 2829 . . . . . . . . . . . . . . . . . . . 20 (𝑐𝐷𝑐 ∈ (𝑊 𝑥𝑊 ran (𝑇𝑥)))
971, 2, 3, 4, 5, 6efgs1 19517 . . . . . . . . . . . . . . . . . . . 20 (𝑐𝐷 → ⟨“𝑐”⟩ ∈ dom 𝑆)
9896, 97sylbir 234 . . . . . . . . . . . . . . . . . . 19 (𝑐 ∈ (𝑊 𝑥𝑊 ran (𝑇𝑥)) → ⟨“𝑐”⟩ ∈ dom 𝑆)
9995, 98sylbir 234 . . . . . . . . . . . . . . . . . 18 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → ⟨“𝑐”⟩ ∈ dom 𝑆)
1001, 2, 3, 4, 5, 6efgsval 19513 . . . . . . . . . . . . . . . . . 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 14494 . . . . . . . . . . . . . . . . . . . . 21 (♯‘⟨“𝑐”⟩) = 1
103102oveq1i 7367 . . . . . . . . . . . . . . . . . . . 20 ((♯‘⟨“𝑐”⟩) − 1) = (1 − 1)
104 1m1e0 12225 . . . . . . . . . . . . . . . . . . . 20 (1 − 1) = 0
105103, 104eqtri 2764 . . . . . . . . . . . . . . . . . . 19 ((♯‘⟨“𝑐”⟩) − 1) = 0
106105fveq2i 6845 . . . . . . . . . . . . . . . . . 18 (⟨“𝑐”⟩‘((♯‘⟨“𝑐”⟩) − 1)) = (⟨“𝑐”⟩‘0)
107106a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → (⟨“𝑐”⟩‘((♯‘⟨“𝑐”⟩) − 1)) = (⟨“𝑐”⟩‘0))
108 s1fv 14498 . . . . . . . . . . . . . . . . . 18 (𝑐𝑊 → (⟨“𝑐”⟩‘0) = 𝑐)
109108adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → (⟨“𝑐”⟩‘0) = 𝑐)
110101, 107, 1093eqtrd 2780 . . . . . . . . . . . . . . . 16 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → (𝑆‘⟨“𝑐”⟩) = 𝑐)
111 fnfvelrn 7031 . . . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . . 15 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → 𝑐 ∈ ran 𝑆)
114113ex 413 . . . . . . . . . . . . . 14 (𝑐𝑊 → (¬ 𝑐 𝑥𝑊 ran (𝑇𝑥) → 𝑐 ∈ ran 𝑆))
1151143ad2ant2 1134 . . . . . . . . . . . . 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 4032 . . . . . . . . . . 11 ((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑐𝑊 ∣ (♯‘𝑐) = 𝑑} ⊆ ran 𝑆)
11843, 117eqsstrid 3992 . . . . . . . . . 10 ((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑} ⊆ ran 𝑆)
11941, 118unssd 4146 . . . . . . . . 9 ((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑}) ⊆ ran 𝑆)
120119ex 413 . . . . . . . 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 12562 . . . . . . . . . . . . 13 (((♯‘𝑎) ∈ ℕ0𝑑 ∈ ℕ0) → ((♯‘𝑎) ≤ 𝑑 ↔ (♯‘𝑎) < (𝑑 + 1)))
12321, 121, 122syl2anr 597 . . . . . . . . . . . 12 ((𝑑 ∈ ℕ0𝑎𝑊) → ((♯‘𝑎) ≤ 𝑑 ↔ (♯‘𝑎) < (𝑑 + 1)))
12421nn0red 12474 . . . . . . . . . . . . 13 (𝑎𝑊 → (♯‘𝑎) ∈ ℝ)
125 nn0re 12422 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ0𝑑 ∈ ℝ)
126 leloe 11241 . . . . . . . . . . . . 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 280 . . . . . . . . . . 11 ((𝑑 ∈ ℕ0𝑎𝑊) → ((♯‘𝑎) < (𝑑 + 1) ↔ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)))
129128rabbidva 3414 . . . . . . . . . 10 (𝑑 ∈ ℕ0 → {𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} = {𝑎𝑊 ∣ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)})
130 unrab 4265 . . . . . . . . . 10 ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑}) = {𝑎𝑊 ∣ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)}
131129, 130eqtr4di 2794 . . . . . . . . 9 (𝑑 ∈ ℕ0 → {𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} = ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑}))
132131sseq1d 3975 . . . . . . . 8 (𝑑 ∈ ℕ0 → ({𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆 ↔ ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑}) ⊆ ran 𝑆))
133120, 132sylibrd 258 . . . . . . 7 (𝑑 ∈ ℕ0 → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆 → {𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆))
13430, 33, 36, 39, 40, 133nn0ind 12598 . . . . . 6 (((♯‘𝑐) + 1) ∈ ℕ0 → {𝑎𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)} ⊆ ran 𝑆)
13517, 18, 1343syl 18 . . . . 5 (𝑐𝑊 → {𝑎𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)} ⊆ ran 𝑆)
136 fveq2 6842 . . . . . . 7 (𝑎 = 𝑐 → (♯‘𝑎) = (♯‘𝑐))
137136breq1d 5115 . . . . . 6 (𝑎 = 𝑐 → ((♯‘𝑎) < ((♯‘𝑐) + 1) ↔ (♯‘𝑐) < ((♯‘𝑐) + 1)))
138 id 22 . . . . . 6 (𝑐𝑊𝑐𝑊)
13917nn0red 12474 . . . . . . 7 (𝑐𝑊 → (♯‘𝑐) ∈ ℝ)
140139ltp1d 12085 . . . . . 6 (𝑐𝑊 → (♯‘𝑐) < ((♯‘𝑐) + 1))
141137, 138, 140elrabd 3647 . . . . 5 (𝑐𝑊𝑐 ∈ {𝑎𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)})
142135, 141sseldd 3945 . . . 4 (𝑐𝑊𝑐 ∈ ran 𝑆)
143142ssriv 3948 . . 3 𝑊 ⊆ ran 𝑆
14412, 143eqssi 3960 . 2 ran 𝑆 = 𝑊
145 dffo2 6760 . 2 (𝑆:dom 𝑆onto𝑊 ↔ (𝑆:dom 𝑆𝑊 ∧ ran 𝑆 = 𝑊))
14610, 144, 145mpbir2an 709 1 𝑆:dom 𝑆onto𝑊
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  {crab 3407  cdif 3907  cun 3908  wss 3910  c0 4282  {csn 4586  cop 4592  cotp 4594   ciun 4954   class class class wbr 5105  cmpt 5188   I cid 5530   × cxp 5631  dom cdm 5633  ran crn 5634   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  (class class class)co 7357  cmpo 7359  1oc1o 8405  2oc2o 8406  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189  cle 11190  cmin 11385  2c2 12208  0cn0 12413  +crp 12915  ...cfz 13424  ..^cfzo 13567  chash 14230  Word cword 14402   ++ cconcat 14458  ⟨“cs1 14483   splice csplice 14637  ⟨“cs2 14730   ~FG cefg 19488
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-splice 14638  df-s2 14737
This theorem is referenced by:  efgredlemc  19527  efgrelexlemb  19532  efgredeu  19534  efgred2  19535
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