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Theorem efgsfo 18794
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 18784 . . 3 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊
87fdmi 6517 . . . 4 dom 𝑆 = {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}
98feq2i 6499 . . 3 (𝑆:dom 𝑆𝑊𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊)
107, 9mpbir 232 . 2 𝑆:dom 𝑆𝑊
11 frn 6513 . . . 4 (𝑆:dom 𝑆𝑊 → ran 𝑆𝑊)
1210, 11ax-mp 5 . . 3 ran 𝑆𝑊
13 fviss 6734 . . . . . . . . 9 ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
141, 13eqsstri 3998 . . . . . . . 8 𝑊 ⊆ Word (𝐼 × 2o)
1514sseli 3960 . . . . . . 7 (𝑐𝑊𝑐 ∈ Word (𝐼 × 2o))
16 lencl 13871 . . . . . . 7 (𝑐 ∈ Word (𝐼 × 2o) → (♯‘𝑐) ∈ ℕ0)
1715, 16syl 17 . . . . . 6 (𝑐𝑊 → (♯‘𝑐) ∈ ℕ0)
18 peano2nn0 11925 . . . . . 6 ((♯‘𝑐) ∈ ℕ0 → ((♯‘𝑐) + 1) ∈ ℕ0)
1914sseli 3960 . . . . . . . . . . . 12 (𝑎𝑊𝑎 ∈ Word (𝐼 × 2o))
20 lencl 13871 . . . . . . . . . . . 12 (𝑎 ∈ Word (𝐼 × 2o) → (♯‘𝑎) ∈ ℕ0)
2119, 20syl 17 . . . . . . . . . . 11 (𝑎𝑊 → (♯‘𝑎) ∈ ℕ0)
22 nn0nlt0 11911 . . . . . . . . . . . 12 ((♯‘𝑎) ∈ ℕ0 → ¬ (♯‘𝑎) < 0)
23 breq2 5061 . . . . . . . . . . . . 13 (𝑏 = 0 → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < 0))
2423notbid 319 . . . . . . . . . . . 12 (𝑏 = 0 → (¬ (♯‘𝑎) < 𝑏 ↔ ¬ (♯‘𝑎) < 0))
2522, 24syl5ibr 247 . . . . . . . . . . 11 (𝑏 = 0 → ((♯‘𝑎) ∈ ℕ0 → ¬ (♯‘𝑎) < 𝑏))
2621, 25syl5 34 . . . . . . . . . 10 (𝑏 = 0 → (𝑎𝑊 → ¬ (♯‘𝑎) < 𝑏))
2726ralrimiv 3178 . . . . . . . . 9 (𝑏 = 0 → ∀𝑎𝑊 ¬ (♯‘𝑎) < 𝑏)
28 rabeq0 4335 . . . . . . . . 9 ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} = ∅ ↔ ∀𝑎𝑊 ¬ (♯‘𝑎) < 𝑏)
2927, 28sylibr 235 . . . . . . . 8 (𝑏 = 0 → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} = ∅)
3029sseq1d 3995 . . . . . . 7 (𝑏 = 0 → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ ∅ ⊆ ran 𝑆))
31 breq2 5061 . . . . . . . . 9 (𝑏 = 𝑑 → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < 𝑑))
3231rabbidv 3478 . . . . . . . 8 (𝑏 = 𝑑 → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑})
3332sseq1d 3995 . . . . . . 7 (𝑏 = 𝑑 → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆))
34 breq2 5061 . . . . . . . . 9 (𝑏 = (𝑑 + 1) → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < (𝑑 + 1)))
3534rabbidv 3478 . . . . . . . 8 (𝑏 = (𝑑 + 1) → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)})
3635sseq1d 3995 . . . . . . 7 (𝑏 = (𝑑 + 1) → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆))
37 breq2 5061 . . . . . . . . 9 (𝑏 = ((♯‘𝑐) + 1) → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < ((♯‘𝑐) + 1)))
3837rabbidv 3478 . . . . . . . 8 (𝑏 = ((♯‘𝑐) + 1) → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)})
3938sseq1d 3995 . . . . . . 7 (𝑏 = ((♯‘𝑐) + 1) → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)} ⊆ ran 𝑆))
40 0ss 4347 . . . . . . 7 ∅ ⊆ ran 𝑆
41 simpr 485 . . . . . . . . . 10 ((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆)
42 fveqeq2 6672 . . . . . . . . . . . 12 (𝑎 = 𝑐 → ((♯‘𝑎) = 𝑑 ↔ (♯‘𝑐) = 𝑑))
4342cbvrabv 3489 . . . . . . . . . . 11 {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑} = {𝑐𝑊 ∣ (♯‘𝑐) = 𝑑}
44 eliun 4914 . . . . . . . . . . . . . . 15 (𝑐 𝑥𝑊 ran (𝑇𝑥) ↔ ∃𝑥𝑊 𝑐 ∈ ran (𝑇𝑥))
45 fveq2 6663 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑏 → (𝑇𝑥) = (𝑇𝑏))
4645rneqd 5801 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑏 → ran (𝑇𝑥) = ran (𝑇𝑏))
4746eleq2d 2895 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑏 → (𝑐 ∈ ran (𝑇𝑥) ↔ 𝑐 ∈ ran (𝑇𝑏)))
4847cbvrexvw 3448 . . . . . . . . . . . . . . 15 (∃𝑥𝑊 𝑐 ∈ ran (𝑇𝑥) ↔ ∃𝑏𝑊 𝑐 ∈ ran (𝑇𝑏))
4944, 48bitri 276 . . . . . . . . . . . . . 14 (𝑐 𝑥𝑊 ran (𝑇𝑥) ↔ ∃𝑏𝑊 𝑐 ∈ ran (𝑇𝑏))
50 simpl1r 1217 . . . . . . . . . . . . . . . . . 18 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆)
51 fveq2 6663 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑏 → (♯‘𝑎) = (♯‘𝑏))
5251breq1d 5067 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑏 → ((♯‘𝑎) < 𝑑 ↔ (♯‘𝑏) < 𝑑))
53 simprl 767 . . . . . . . . . . . . . . . . . . 19 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → 𝑏𝑊)
5414, 53sseldi 3962 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → 𝑏 ∈ Word (𝐼 × 2o))
55 lencl 13871 . . . . . . . . . . . . . . . . . . . . . . 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 11944 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → (♯‘𝑏) ∈ ℝ)
58 2rp 12382 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℝ+
59 ltaddrp 12414 . . . . . . . . . . . . . . . . . . . . 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 18781 . . . . . . . . . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → (♯‘𝑐) = 𝑑)
6462, 63eqtr3d 2855 . . . . . . . . . . . . . . . . . . . 20 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → ((♯‘𝑏) + 2) = 𝑑)
6560, 64breqtrd 5083 . . . . . . . . . . . . . . . . . . 19 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → (♯‘𝑏) < 𝑑)
6652, 53, 65elrabd 3679 . . . . . . . . . . . . . . . . . 18 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → 𝑏 ∈ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑})
6750, 66sseldd 3965 . . . . . . . . . . . . . . . . 17 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → 𝑏 ∈ ran 𝑆)
68 ffn 6507 . . . . . . . . . . . . . . . . . . 19 (𝑆:dom 𝑆𝑊𝑆 Fn dom 𝑆)
6910, 68ax-mp 5 . . . . . . . . . . . . . . . . . 18 𝑆 Fn dom 𝑆
70 fvelrnb 6719 . . . . . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . . . . 16 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → ∃𝑜 ∈ dom 𝑆(𝑆𝑜) = 𝑏)
73 simprrl 777 . . . . . . . . . . . . . . . . . . . 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 18785 . . . . . . . . . . . . . . . . . . . . 21 (𝑜 ∈ dom 𝑆 ↔ (𝑜 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑜‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑜))(𝑜𝑖) ∈ ran (𝑇‘(𝑜‘(𝑖 − 1)))))
7574simp1bi 1137 . . . . . . . . . . . . . . . . . . . 20 (𝑜 ∈ dom 𝑆𝑜 ∈ (Word 𝑊 ∖ {∅}))
76 eldifi 4100 . . . . . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . . . . . 19 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → 𝑐𝑊)
79 simprlr 776 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → 𝑐 ∈ ran (𝑇𝑏))
80 simprrr 778 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → (𝑆𝑜) = 𝑏)
8180fveq2d 6667 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏𝑊𝑐 ∈ ran (𝑇𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆𝑜) = 𝑏))) → (𝑇‘(𝑆𝑜)) = (𝑇𝑏))
8281rneqd 5801 . . . . . . . . . . . . . . . . . . . . 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 2913 . . . . . . . . . . . . . . . . . . . 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 18792 . . . . . . . . . . . . . . . . . . . 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 18788 . . . . . . . . . . . . . . . . . . 19 ((𝑜 ∈ Word 𝑊𝑐𝑊 ∧ (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) = 𝑐)
8777, 78, 85, 86syl3anc 1363 . . . . . . . . . . . . . . . . . 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 6840 . . . . . . . . . . . . . . . . . . 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 2911 . . . . . . . . . . . . . . . . 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 3288 . . . . . . . . . . . . . . 15 ((((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏𝑊𝑐 ∈ ran (𝑇𝑏))) → 𝑐 ∈ ran 𝑆)
9392rexlimdvaa 3282 . . . . . . . . . . . . . 14 (((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) → (∃𝑏𝑊 𝑐 ∈ ran (𝑇𝑏) → 𝑐 ∈ ran 𝑆))
9449, 93syl5bi 243 . . . . . . . . . . . . 13 (((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) → (𝑐 𝑥𝑊 ran (𝑇𝑥) → 𝑐 ∈ ran 𝑆))
95 eldif 3943 . . . . . . . . . . . . . . . . . . 19 (𝑐 ∈ (𝑊 𝑥𝑊 ran (𝑇𝑥)) ↔ (𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)))
965eleq2i 2901 . . . . . . . . . . . . . . . . . . . 20 (𝑐𝐷𝑐 ∈ (𝑊 𝑥𝑊 ran (𝑇𝑥)))
971, 2, 3, 4, 5, 6efgs1 18790 . . . . . . . . . . . . . . . . . . . 20 (𝑐𝐷 → ⟨“𝑐”⟩ ∈ dom 𝑆)
9896, 97sylbir 236 . . . . . . . . . . . . . . . . . . 19 (𝑐 ∈ (𝑊 𝑥𝑊 ran (𝑇𝑥)) → ⟨“𝑐”⟩ ∈ dom 𝑆)
9995, 98sylbir 236 . . . . . . . . . . . . . . . . . 18 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → ⟨“𝑐”⟩ ∈ dom 𝑆)
1001, 2, 3, 4, 5, 6efgsval 18786 . . . . . . . . . . . . . . . . . 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 13948 . . . . . . . . . . . . . . . . . . . . 21 (♯‘⟨“𝑐”⟩) = 1
103102oveq1i 7155 . . . . . . . . . . . . . . . . . . . 20 ((♯‘⟨“𝑐”⟩) − 1) = (1 − 1)
104 1m1e0 11697 . . . . . . . . . . . . . . . . . . . 20 (1 − 1) = 0
105103, 104eqtri 2841 . . . . . . . . . . . . . . . . . . 19 ((♯‘⟨“𝑐”⟩) − 1) = 0
106105fveq2i 6666 . . . . . . . . . . . . . . . . . 18 (⟨“𝑐”⟩‘((♯‘⟨“𝑐”⟩) − 1)) = (⟨“𝑐”⟩‘0)
107106a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → (⟨“𝑐”⟩‘((♯‘⟨“𝑐”⟩) − 1)) = (⟨“𝑐”⟩‘0))
108 s1fv 13952 . . . . . . . . . . . . . . . . . 18 (𝑐𝑊 → (⟨“𝑐”⟩‘0) = 𝑐)
109108adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → (⟨“𝑐”⟩‘0) = 𝑐)
110101, 107, 1093eqtrd 2857 . . . . . . . . . . . . . . . 16 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → (𝑆‘⟨“𝑐”⟩) = 𝑐)
111 fnfvelrn 6840 . . . . . . . . . . . . . . . . 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 2911 . . . . . . . . . . . . . . 15 ((𝑐𝑊 ∧ ¬ 𝑐 𝑥𝑊 ran (𝑇𝑥)) → 𝑐 ∈ ran 𝑆)
114113ex 413 . . . . . . . . . . . . . 14 (𝑐𝑊 → (¬ 𝑐 𝑥𝑊 ran (𝑇𝑥) → 𝑐 ∈ ran 𝑆))
1151143ad2ant2 1126 . . . . . . . . . . . . 13 (((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) → (¬ 𝑐 𝑥𝑊 ran (𝑇𝑥) → 𝑐 ∈ ran 𝑆))
11694, 115pm2.61d 180 . . . . . . . . . . . 12 (((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐𝑊 ∧ (♯‘𝑐) = 𝑑) → 𝑐 ∈ ran 𝑆)
117116rabssdv 4048 . . . . . . . . . . 11 ((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑐𝑊 ∣ (♯‘𝑐) = 𝑑} ⊆ ran 𝑆)
11843, 117eqsstrid 4012 . . . . . . . . . 10 ((𝑑 ∈ ℕ0 ∧ {𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑} ⊆ ran 𝑆)
11941, 118unssd 4159 . . . . . . . . 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 12029 . . . . . . . . . . . . 13 (((♯‘𝑎) ∈ ℕ0𝑑 ∈ ℕ0) → ((♯‘𝑎) ≤ 𝑑 ↔ (♯‘𝑎) < (𝑑 + 1)))
12321, 121, 122syl2anr 596 . . . . . . . . . . . 12 ((𝑑 ∈ ℕ0𝑎𝑊) → ((♯‘𝑎) ≤ 𝑑 ↔ (♯‘𝑎) < (𝑑 + 1)))
12421nn0red 11944 . . . . . . . . . . . . 13 (𝑎𝑊 → (♯‘𝑎) ∈ ℝ)
125 nn0re 11894 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ0𝑑 ∈ ℝ)
126 leloe 10715 . . . . . . . . . . . . 13 (((♯‘𝑎) ∈ ℝ ∧ 𝑑 ∈ ℝ) → ((♯‘𝑎) ≤ 𝑑 ↔ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)))
127124, 125, 126syl2anr 596 . . . . . . . . . . . 12 ((𝑑 ∈ ℕ0𝑎𝑊) → ((♯‘𝑎) ≤ 𝑑 ↔ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)))
128123, 127bitr3d 282 . . . . . . . . . . 11 ((𝑑 ∈ ℕ0𝑎𝑊) → ((♯‘𝑎) < (𝑑 + 1) ↔ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)))
129128rabbidva 3476 . . . . . . . . . 10 (𝑑 ∈ ℕ0 → {𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} = {𝑎𝑊 ∣ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)})
130 unrab 4271 . . . . . . . . . 10 ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑}) = {𝑎𝑊 ∣ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)}
131129, 130syl6eqr 2871 . . . . . . . . 9 (𝑑 ∈ ℕ0 → {𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} = ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑}))
132131sseq1d 3995 . . . . . . . 8 (𝑑 ∈ ℕ0 → ({𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆 ↔ ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎𝑊 ∣ (♯‘𝑎) = 𝑑}) ⊆ ran 𝑆))
133120, 132sylibrd 260 . . . . . . 7 (𝑑 ∈ ℕ0 → ({𝑎𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆 → {𝑎𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆))
13430, 33, 36, 39, 40, 133nn0ind 12065 . . . . . 6 (((♯‘𝑐) + 1) ∈ ℕ0 → {𝑎𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)} ⊆ ran 𝑆)
13517, 18, 1343syl 18 . . . . 5 (𝑐𝑊 → {𝑎𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)} ⊆ ran 𝑆)
136 fveq2 6663 . . . . . . 7 (𝑎 = 𝑐 → (♯‘𝑎) = (♯‘𝑐))
137136breq1d 5067 . . . . . 6 (𝑎 = 𝑐 → ((♯‘𝑎) < ((♯‘𝑐) + 1) ↔ (♯‘𝑐) < ((♯‘𝑐) + 1)))
138 id 22 . . . . . 6 (𝑐𝑊𝑐𝑊)
13917nn0red 11944 . . . . . . 7 (𝑐𝑊 → (♯‘𝑐) ∈ ℝ)
140139ltp1d 11558 . . . . . 6 (𝑐𝑊 → (♯‘𝑐) < ((♯‘𝑐) + 1))
141137, 138, 140elrabd 3679 . . . . 5 (𝑐𝑊𝑐 ∈ {𝑎𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)})
142135, 141sseldd 3965 . . . 4 (𝑐𝑊𝑐 ∈ ran 𝑆)
143142ssriv 3968 . . 3 𝑊 ⊆ ran 𝑆
14412, 143eqssi 3980 . 2 ran 𝑆 = 𝑊
145 dffo2 6587 . 2 (𝑆:dom 𝑆onto𝑊 ↔ (𝑆:dom 𝑆𝑊 ∧ ran 𝑆 = 𝑊))
14610, 144, 145mpbir2an 707 1 𝑆:dom 𝑆onto𝑊
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  cdif 3930  cun 3931  wss 3933  c0 4288  {csn 4557  cop 4563  cotp 4565   ciun 4910   class class class wbr 5057  cmpt 5137   I cid 5452   × cxp 5546  dom cdm 5548  ran crn 5549   Fn wfn 6343  wf 6344  ontowfo 6346  cfv 6348  (class class class)co 7145  cmpo 7147  1oc1o 8084  2oc2o 8085  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   < clt 10663  cle 10664  cmin 10858  2c2 11680  0cn0 11885  +crp 12377  ...cfz 12880  ..^cfzo 13021  chash 13678  Word cword 13849   ++ cconcat 13910  ⟨“cs1 13937   splice csplice 14099  ⟨“cs2 14191   ~FG cefg 18761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-ot 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-hash 13679  df-word 13850  df-concat 13911  df-s1 13938  df-substr 13991  df-pfx 14021  df-splice 14100  df-s2 14198
This theorem is referenced by:  efgredlemc  18800  efgrelexlemb  18805  efgredeu  18807  efgred2  18808
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