Theorem efgsfo 19535

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	⊢ ∼ = ( ~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

Ref	Expression
efgsfo	⊢ 𝑆:dom 𝑆–onto→𝑊

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.

Step	Hyp	Ref	Expression
1		efgval.w	. . . 4 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
2		efgval.r	. . . 4 ⊢ ∼ = ( ~FG ‘𝐼)
3		efgval2.m	. . . 4 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
4		efgval2.t	. . . 4 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
5		efgred.d	. . . 4 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
6		efgred.s	. . . 4 ⊢ 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	efgsf 19525	. . 3 ⊢ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊
8	7	fdmi 6685	. . . 4 ⊢ dom 𝑆 = {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}
9	8	feq2i 6665	. . 3 ⊢ (𝑆:dom 𝑆⟶𝑊 ↔ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊)
10	7, 9	mpbir 230	. 2 ⊢ 𝑆:dom 𝑆⟶𝑊
11		frn 6680	. . . 4 ⊢ (𝑆:dom 𝑆⟶𝑊 → ran 𝑆 ⊆ 𝑊)
12	10, 11	ax-mp 5	. . 3 ⊢ ran 𝑆 ⊆ 𝑊
13		fviss 6923	. . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
14	1, 13	eqsstri 3981	. . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
15	14	sseli 3943	. . . . . . 7 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ Word (𝐼 × 2o))
16		lencl 14433	. . . . . . 7 ⊢ (𝑐 ∈ Word (𝐼 × 2o) → (♯‘𝑐) ∈ ℕ0)
17	15, 16	syl 17	. . . . . 6 ⊢ (𝑐 ∈ 𝑊 → (♯‘𝑐) ∈ ℕ0)
18		peano2nn0 12462	. . . . . 6 ⊢ ((♯‘𝑐) ∈ ℕ0 → ((♯‘𝑐) + 1) ∈ ℕ0)
19	14	sseli 3943	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑊 → 𝑎 ∈ Word (𝐼 × 2o))
20		lencl 14433	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ Word (𝐼 × 2o) → (♯‘𝑎) ∈ ℕ0)
21	19, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝑊 → (♯‘𝑎) ∈ ℕ0)
22		nn0nlt0 12448	. . . . . . . . . . . 12 ⊢ ((♯‘𝑎) ∈ ℕ0 → ¬ (♯‘𝑎) < 0)
23		breq2 5114	. . . . . . . . . . . . 13 ⊢ (𝑏 = 0 → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < 0))
24	23	notbid 317	. . . . . . . . . . . 12 ⊢ (𝑏 = 0 → (¬ (♯‘𝑎) < 𝑏 ↔ ¬ (♯‘𝑎) < 0))
25	22, 24	imbitrrid 245	. . . . . . . . . . 11 ⊢ (𝑏 = 0 → ((♯‘𝑎) ∈ ℕ0 → ¬ (♯‘𝑎) < 𝑏))
26	21, 25	syl5 34	. . . . . . . . . 10 ⊢ (𝑏 = 0 → (𝑎 ∈ 𝑊 → ¬ (♯‘𝑎) < 𝑏))
27	26	ralrimiv 3138	. . . . . . . . 9 ⊢ (𝑏 = 0 → ∀𝑎 ∈ 𝑊 ¬ (♯‘𝑎) < 𝑏)
28		rabeq0 4349	. . . . . . . . 9 ⊢ ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} = ∅ ↔ ∀𝑎 ∈ 𝑊 ¬ (♯‘𝑎) < 𝑏)
29	27, 28	sylibr 233	. . . . . . . 8 ⊢ (𝑏 = 0 → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} = ∅)
30	29	sseq1d 3978	. . . . . . 7 ⊢ (𝑏 = 0 → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ ∅ ⊆ ran 𝑆))
31		breq2 5114	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < 𝑑))
32	31	rabbidv 3413	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑})
33	32	sseq1d 3978	. . . . . . 7 ⊢ (𝑏 = 𝑑 → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆))
34		breq2 5114	. . . . . . . . 9 ⊢ (𝑏 = (𝑑 + 1) → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < (𝑑 + 1)))
35	34	rabbidv 3413	. . . . . . . 8 ⊢ (𝑏 = (𝑑 + 1) → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)})
36	35	sseq1d 3978	. . . . . . 7 ⊢ (𝑏 = (𝑑 + 1) → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆))
37		breq2 5114	. . . . . . . . 9 ⊢ (𝑏 = ((♯‘𝑐) + 1) → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < ((♯‘𝑐) + 1)))
38	37	rabbidv 3413	. . . . . . . 8 ⊢ (𝑏 = ((♯‘𝑐) + 1) → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)})
39	38	sseq1d 3978	. . . . . . 7 ⊢ (𝑏 = ((♯‘𝑐) + 1) → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)} ⊆ ran 𝑆))
40		0ss 4361	. . . . . . 7 ⊢ ∅ ⊆ ran 𝑆
41		simpr 485	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆)
42		fveqeq2 6856	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → ((♯‘𝑎) = 𝑑 ↔ (♯‘𝑐) = 𝑑))
43	42	cbvrabv 3415	. . . . . . . . . . 11 ⊢ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) = 𝑑} = {𝑐 ∈ 𝑊 ∣ (♯‘𝑐) = 𝑑}
44		eliun 4963	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ↔ ∃𝑥 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑥))
45		fveq2 6847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑏 → (𝑇‘𝑥) = (𝑇‘𝑏))
46	45	rneqd 5898	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑏 → ran (𝑇‘𝑥) = ran (𝑇‘𝑏))
47	46	eleq2d 2818	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑏 → (𝑐 ∈ ran (𝑇‘𝑥) ↔ 𝑐 ∈ ran (𝑇‘𝑏)))
48	47	cbvrexvw 3224	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑥) ↔ ∃𝑏 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑏))
49	44, 48	bitri 274	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ↔ ∃𝑏 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑏))
50		simpl1r 1225	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆)
51		fveq2 6847	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑏 → (♯‘𝑎) = (♯‘𝑏))
52	51	breq1d 5120	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑏 → ((♯‘𝑎) < 𝑑 ↔ (♯‘𝑏) < 𝑑))
53		simprl 769	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → 𝑏 ∈ 𝑊)
54	14, 53	sselid 3945	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → 𝑏 ∈ Word (𝐼 × 2o))
55		lencl 14433	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 ∈ Word (𝐼 × 2o) → (♯‘𝑏) ∈ ℕ0)
56	54, 55	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (♯‘𝑏) ∈ ℕ0)
57	56	nn0red 12483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (♯‘𝑏) ∈ ℝ)
58		2rp 12929	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ+
59		ltaddrp 12961	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝑏) ∈ ℝ ∧ 2 ∈ ℝ+) → (♯‘𝑏) < ((♯‘𝑏) + 2))
60	57, 58, 59	sylancl 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (♯‘𝑏) < ((♯‘𝑏) + 2))
61	1, 2, 3, 4	efgtlen 19522	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏)) → (♯‘𝑐) = ((♯‘𝑏) + 2))
62	61	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (♯‘𝑐) = ((♯‘𝑏) + 2))
63		simpl3 1193	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (♯‘𝑐) = 𝑑)
64	62, 63	eqtr3d 2773	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → ((♯‘𝑏) + 2) = 𝑑)
65	60, 64	breqtrd 5136	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (♯‘𝑏) < 𝑑)
66	52, 53, 65	elrabd 3650	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → 𝑏 ∈ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑})
67	50, 66	sseldd 3948	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → 𝑏 ∈ ran 𝑆)
68		ffn 6673	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆:dom 𝑆⟶𝑊 → 𝑆 Fn dom 𝑆)
69	10, 68	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑆 Fn dom 𝑆
70		fvelrnb 6908	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 Fn dom 𝑆 → (𝑏 ∈ ran 𝑆 ↔ ∃𝑜 ∈ dom 𝑆(𝑆‘𝑜) = 𝑏))
71	69, 70	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ ran 𝑆 ↔ ∃𝑜 ∈ dom 𝑆(𝑆‘𝑜) = 𝑏)
72	67, 71	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → ∃𝑜 ∈ dom 𝑆(𝑆‘𝑜) = 𝑏)
73		simprrl 779	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆‘𝑜) = 𝑏))) → 𝑜 ∈ dom 𝑆)
74	1, 2, 3, 4, 5, 6	efgsdm 19526	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑜 ∈ dom 𝑆 ↔ (𝑜 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑜‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑜))(𝑜‘𝑖) ∈ ran (𝑇‘(𝑜‘(𝑖 − 1)))))
75	74	simp1bi 1145	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑜 ∈ dom 𝑆 → 𝑜 ∈ (Word 𝑊 ∖ {∅}))
76		eldifi 4091	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑜 ∈ (Word 𝑊 ∖ {∅}) → 𝑜 ∈ Word 𝑊)
77	73, 75, 76	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆‘𝑜) = 𝑏))) → 𝑜 ∈ Word 𝑊)
78		simpl2 1192	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆‘𝑜) = 𝑏))) → 𝑐 ∈ 𝑊)
79		simprlr 778	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆‘𝑜) = 𝑏))) → 𝑐 ∈ ran (𝑇‘𝑏))
80		simprrr 780	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆‘𝑜) = 𝑏))) → (𝑆‘𝑜) = 𝑏)
81	80	fveq2d 6851	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆‘𝑜) = 𝑏))) → (𝑇‘(𝑆‘𝑜)) = (𝑇‘𝑏))
82	81	rneqd 5898	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆‘𝑜) = 𝑏))) → ran (𝑇‘(𝑆‘𝑜)) = ran (𝑇‘𝑏))
83	79, 82	eleqtrrd 2835	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆‘𝑜) = 𝑏))) → 𝑐 ∈ ran (𝑇‘(𝑆‘𝑜)))
84	1, 2, 3, 4, 5, 6	efgsp1 19533	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑜 ∈ dom 𝑆 ∧ 𝑐 ∈ ran (𝑇‘(𝑆‘𝑜))) → (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆)
85	73, 83, 84	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆‘𝑜) = 𝑏))) → (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆)
86	1, 2, 3, 4, 5, 6	efgsval2 19529	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑜 ∈ Word 𝑊 ∧ 𝑐 ∈ 𝑊 ∧ (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) = 𝑐)
87	77, 78, 85, 86	syl3anc 1371	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆‘𝑜) = 𝑏))) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) = 𝑐)
88		fnfvelrn 7036	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 Fn dom 𝑆 ∧ (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) ∈ ran 𝑆)
89	69, 85, 88	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆‘𝑜) = 𝑏))) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) ∈ ran 𝑆)
90	87, 89	eqeltrrd 2833	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ ((𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏)) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆‘𝑜) = 𝑏))) → 𝑐 ∈ ran 𝑆)
91	90	anassrs 468	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) ∧ (𝑜 ∈ dom 𝑆 ∧ (𝑆‘𝑜) = 𝑏)) → 𝑐 ∈ ran 𝑆)
92	72, 91	rexlimddv 3154	. . . . . . . . . . . . . . 15 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → 𝑐 ∈ ran 𝑆)
93	92	rexlimdvaa 3149	. . . . . . . . . . . . . 14 ⊢ (((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) → (∃𝑏 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑏) → 𝑐 ∈ ran 𝑆))
94	49, 93	biimtrid 241	. . . . . . . . . . . . 13 ⊢ (((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) → (𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) → 𝑐 ∈ ran 𝑆))
95		eldif 3923	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) ↔ (𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)))
96	5	eleq2i 2824	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ 𝐷 ↔ 𝑐 ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)))
97	1, 2, 3, 4, 5, 6	efgs1 19531	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ 𝐷 → ⟨“𝑐”⟩ ∈ dom 𝑆)
98	96, 97	sylbir 234	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → ⟨“𝑐”⟩ ∈ dom 𝑆)
99	95, 98	sylbir 234	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → ⟨“𝑐”⟩ ∈ dom 𝑆)
100	1, 2, 3, 4, 5, 6	efgsval 19527	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨“𝑐”⟩ ∈ dom 𝑆 → (𝑆‘⟨“𝑐”⟩) = (⟨“𝑐”⟩‘((♯‘⟨“𝑐”⟩) − 1)))
101	99, 100	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (𝑆‘⟨“𝑐”⟩) = (⟨“𝑐”⟩‘((♯‘⟨“𝑐”⟩) − 1)))
102		s1len 14506	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (♯‘⟨“𝑐”⟩) = 1
103	102	oveq1i 7372	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘⟨“𝑐”⟩) − 1) = (1 − 1)
104		1m1e0 12234	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 − 1) = 0
105	103, 104	eqtri 2759	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘⟨“𝑐”⟩) − 1) = 0
106	105	fveq2i 6850	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨“𝑐”⟩‘((♯‘⟨“𝑐”⟩) − 1)) = (⟨“𝑐”⟩‘0)
107	106	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (⟨“𝑐”⟩‘((♯‘⟨“𝑐”⟩) − 1)) = (⟨“𝑐”⟩‘0))
108		s1fv 14510	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 ∈ 𝑊 → (⟨“𝑐”⟩‘0) = 𝑐)
109	108	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (⟨“𝑐”⟩‘0) = 𝑐)
110	101, 107, 109	3eqtrd 2775	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (𝑆‘⟨“𝑐”⟩) = 𝑐)
111		fnfvelrn 7036	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 Fn dom 𝑆 ∧ ⟨“𝑐”⟩ ∈ dom 𝑆) → (𝑆‘⟨“𝑐”⟩) ∈ ran 𝑆)
112	69, 99, 111	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (𝑆‘⟨“𝑐”⟩) ∈ ran 𝑆)
113	110, 112	eqeltrrd 2833	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → 𝑐 ∈ ran 𝑆)
114	113	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝑊 → (¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) → 𝑐 ∈ ran 𝑆))
115	114	3ad2ant2 1134	. . . . . . . . . . . . 13 ⊢ (((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) → (¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) → 𝑐 ∈ ran 𝑆))
116	94, 115	pm2.61d 179	. . . . . . . . . . . 12 ⊢ (((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) → 𝑐 ∈ ran 𝑆)
117	116	rabssdv 4037	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑐 ∈ 𝑊 ∣ (♯‘𝑐) = 𝑑} ⊆ ran 𝑆)
118	43, 117	eqsstrid 3995	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) = 𝑑} ⊆ ran 𝑆)
119	41, 118	unssd 4151	. . . . . . . . 9 ⊢ ((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) = 𝑑}) ⊆ ran 𝑆)
120	119	ex 413	. . . . . . . 8 ⊢ (𝑑 ∈ ℕ0 → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆 → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) = 𝑑}) ⊆ ran 𝑆))
121		id 22	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ0 → 𝑑 ∈ ℕ0)
122		nn0leltp1 12571	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑎) ∈ ℕ0 ∧ 𝑑 ∈ ℕ0) → ((♯‘𝑎) ≤ 𝑑 ↔ (♯‘𝑎) < (𝑑 + 1)))
123	21, 121, 122	syl2anr 597	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ ℕ0 ∧ 𝑎 ∈ 𝑊) → ((♯‘𝑎) ≤ 𝑑 ↔ (♯‘𝑎) < (𝑑 + 1)))
124	21	nn0red 12483	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑊 → (♯‘𝑎) ∈ ℝ)
125		nn0re 12431	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ0 → 𝑑 ∈ ℝ)
126		leloe 11250	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑎) ∈ ℝ ∧ 𝑑 ∈ ℝ) → ((♯‘𝑎) ≤ 𝑑 ↔ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)))
127	124, 125, 126	syl2anr 597	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ ℕ0 ∧ 𝑎 ∈ 𝑊) → ((♯‘𝑎) ≤ 𝑑 ↔ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)))
128	123, 127	bitr3d 280	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ ℕ0 ∧ 𝑎 ∈ 𝑊) → ((♯‘𝑎) < (𝑑 + 1) ↔ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)))
129	128	rabbidva 3412	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℕ0 → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} = {𝑎 ∈ 𝑊 ∣ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)})
130		unrab 4270	. . . . . . . . . 10 ⊢ ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) = 𝑑}) = {𝑎 ∈ 𝑊 ∣ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)}
131	129, 130	eqtr4di 2789	. . . . . . . . 9 ⊢ (𝑑 ∈ ℕ0 → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} = ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) = 𝑑}))
132	131	sseq1d 3978	. . . . . . . 8 ⊢ (𝑑 ∈ ℕ0 → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆 ↔ ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) = 𝑑}) ⊆ ran 𝑆))
133	120, 132	sylibrd 258	. . . . . . 7 ⊢ (𝑑 ∈ ℕ0 → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆 → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆))
134	30, 33, 36, 39, 40, 133	nn0ind 12607	. . . . . 6 ⊢ (((♯‘𝑐) + 1) ∈ ℕ0 → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)} ⊆ ran 𝑆)
135	17, 18, 134	3syl 18	. . . . 5 ⊢ (𝑐 ∈ 𝑊 → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)} ⊆ ran 𝑆)
136		fveq2 6847	. . . . . . 7 ⊢ (𝑎 = 𝑐 → (♯‘𝑎) = (♯‘𝑐))
137	136	breq1d 5120	. . . . . 6 ⊢ (𝑎 = 𝑐 → ((♯‘𝑎) < ((♯‘𝑐) + 1) ↔ (♯‘𝑐) < ((♯‘𝑐) + 1)))
138		id 22	. . . . . 6 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ 𝑊)
139	17	nn0red 12483	. . . . . . 7 ⊢ (𝑐 ∈ 𝑊 → (♯‘𝑐) ∈ ℝ)
140	139	ltp1d 12094	. . . . . 6 ⊢ (𝑐 ∈ 𝑊 → (♯‘𝑐) < ((♯‘𝑐) + 1))
141	137, 138, 140	elrabd 3650	. . . . 5 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)})
142	135, 141	sseldd 3948	. . . 4 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ ran 𝑆)
143	142	ssriv 3951	. . 3 ⊢ 𝑊 ⊆ ran 𝑆
144	12, 143	eqssi 3963	. 2 ⊢ ran 𝑆 = 𝑊
145		dffo2 6765	. 2 ⊢ (𝑆:dom 𝑆–onto→𝑊 ↔ (𝑆:dom 𝑆⟶𝑊 ∧ ran 𝑆 = 𝑊))
146	10, 144, 145	mpbir2an 709	1 ⊢ 𝑆:dom 𝑆–onto→𝑊

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069  {crab 3405   ∖ cdif 3910   ∪ cun 3911   ⊆ wss 3913  ∅c0 4287  {csn 4591  ⟨cop 4597  ⟨cotp 4599  ∪ ciun 4959   class class class wbr 5110   ↦ cmpt 5193   I cid 5535   × cxp 5636  dom cdm 5638  ran crn 5639   Fn wfn 6496  ⟶wf 6497  –onto→wfo 6499  ‘cfv 6501  (class class class)co 7362   ∈ cmpo 7364  1_oc1o 8410  2_oc2o 8411  ℝcr 11059  0cc0 11060  1c1 11061   + caddc 11063   < clt 11198   ≤ cle 11199   − cmin 11394  2c2 12217  ℕ₀cn0 12422  ℝ⁺crp 12924  ...cfz 13434  ..^cfzo 13577  ♯chash 14240  Word cword 14414   ++ cconcat 14470  ⟨“cs1 14495   splice csplice 14649  ⟨“cs2 14742   ~_FG cefg 19502

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 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137

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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-ot 4600  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9884  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-nn 12163  df-2 12225  df-n0 12423  df-xnn0 12495  df-z 12509  df-uz 12773  df-rp 12925  df-fz 13435  df-fzo 13578  df-hash 14241  df-word 14415  df-concat 14471  df-s1 14496  df-substr 14541  df-pfx 14571  df-splice 14650  df-s2 14749

This theorem is referenced by:  efgredlemc  19541  efgrelexlemb  19546  efgredeu  19548  efgred2  19549