Theorem efgsfo 19260

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 19250	. . 3 ⊢ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊
8	7	fdmi 6596	. . . 4 ⊢ dom 𝑆 = {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}
9	8	feq2i 6576	. . 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 6591	. . . 4 ⊢ (𝑆:dom 𝑆⟶𝑊 → ran 𝑆 ⊆ 𝑊)
12	10, 11	ax-mp 5	. . 3 ⊢ ran 𝑆 ⊆ 𝑊
13		fviss 6827	. . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
14	1, 13	eqsstri 3951	. . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
15	14	sseli 3913	. . . . . . 7 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ Word (𝐼 × 2o))
16		lencl 14164	. . . . . . 7 ⊢ (𝑐 ∈ Word (𝐼 × 2o) → (♯‘𝑐) ∈ ℕ0)
17	15, 16	syl 17	. . . . . 6 ⊢ (𝑐 ∈ 𝑊 → (♯‘𝑐) ∈ ℕ0)
18		peano2nn0 12203	. . . . . 6 ⊢ ((♯‘𝑐) ∈ ℕ0 → ((♯‘𝑐) + 1) ∈ ℕ0)
19	14	sseli 3913	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑊 → 𝑎 ∈ Word (𝐼 × 2o))
20		lencl 14164	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ Word (𝐼 × 2o) → (♯‘𝑎) ∈ ℕ0)
21	19, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝑊 → (♯‘𝑎) ∈ ℕ0)
22		nn0nlt0 12189	. . . . . . . . . . . 12 ⊢ ((♯‘𝑎) ∈ ℕ0 → ¬ (♯‘𝑎) < 0)
23		breq2 5074	. . . . . . . . . . . . 13 ⊢ (𝑏 = 0 → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < 0))
24	23	notbid 317	. . . . . . . . . . . 12 ⊢ (𝑏 = 0 → (¬ (♯‘𝑎) < 𝑏 ↔ ¬ (♯‘𝑎) < 0))
25	22, 24	syl5ibr 245	. . . . . . . . . . 11 ⊢ (𝑏 = 0 → ((♯‘𝑎) ∈ ℕ0 → ¬ (♯‘𝑎) < 𝑏))
26	21, 25	syl5 34	. . . . . . . . . 10 ⊢ (𝑏 = 0 → (𝑎 ∈ 𝑊 → ¬ (♯‘𝑎) < 𝑏))
27	26	ralrimiv 3106	. . . . . . . . 9 ⊢ (𝑏 = 0 → ∀𝑎 ∈ 𝑊 ¬ (♯‘𝑎) < 𝑏)
28		rabeq0 4315	. . . . . . . . 9 ⊢ ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} = ∅ ↔ ∀𝑎 ∈ 𝑊 ¬ (♯‘𝑎) < 𝑏)
29	27, 28	sylibr 233	. . . . . . . 8 ⊢ (𝑏 = 0 → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} = ∅)
30	29	sseq1d 3948	. . . . . . 7 ⊢ (𝑏 = 0 → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ ∅ ⊆ ran 𝑆))
31		breq2 5074	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < 𝑑))
32	31	rabbidv 3404	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑})
33	32	sseq1d 3948	. . . . . . 7 ⊢ (𝑏 = 𝑑 → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆))
34		breq2 5074	. . . . . . . . 9 ⊢ (𝑏 = (𝑑 + 1) → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < (𝑑 + 1)))
35	34	rabbidv 3404	. . . . . . . 8 ⊢ (𝑏 = (𝑑 + 1) → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)})
36	35	sseq1d 3948	. . . . . . 7 ⊢ (𝑏 = (𝑑 + 1) → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆))
37		breq2 5074	. . . . . . . . 9 ⊢ (𝑏 = ((♯‘𝑐) + 1) → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < ((♯‘𝑐) + 1)))
38	37	rabbidv 3404	. . . . . . . 8 ⊢ (𝑏 = ((♯‘𝑐) + 1) → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)})
39	38	sseq1d 3948	. . . . . . 7 ⊢ (𝑏 = ((♯‘𝑐) + 1) → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)} ⊆ ran 𝑆))
40		0ss 4327	. . . . . . 7 ⊢ ∅ ⊆ ran 𝑆
41		simpr 484	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆)
42		fveqeq2 6765	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → ((♯‘𝑎) = 𝑑 ↔ (♯‘𝑐) = 𝑑))
43	42	cbvrabv 3416	. . . . . . . . . . 11 ⊢ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) = 𝑑} = {𝑐 ∈ 𝑊 ∣ (♯‘𝑐) = 𝑑}
44		eliun 4925	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ↔ ∃𝑥 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑥))
45		fveq2 6756	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑏 → (𝑇‘𝑥) = (𝑇‘𝑏))
46	45	rneqd 5836	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑏 → ran (𝑇‘𝑥) = ran (𝑇‘𝑏))
47	46	eleq2d 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑏 → (𝑐 ∈ ran (𝑇‘𝑥) ↔ 𝑐 ∈ ran (𝑇‘𝑏)))
48	47	cbvrexvw 3373	. . . . . . . . . . . . . . 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 1223	. . . . . . . . . . . . . . . . . 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 6756	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑏 → (♯‘𝑎) = (♯‘𝑏))
52	51	breq1d 5080	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑏 → ((♯‘𝑎) < 𝑑 ↔ (♯‘𝑏) < 𝑑))
53		simprl 767	. . . . . . . . . . . . . . . . . . 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 3915	. . . . . . . . . . . . . . . . . . . . . . 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 14164	. . . . . . . . . . . . . . . . . . . . . . 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 12224	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (♯‘𝑏) ∈ ℝ)
58		2rp 12664	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ+
59		ltaddrp 12696	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝑏) ∈ ℝ ∧ 2 ∈ ℝ+) → (♯‘𝑏) < ((♯‘𝑏) + 2))
60	57, 58, 59	sylancl 585	. . . . . . . . . . . . . . . . . . . 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 19247	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏)) → (♯‘𝑐) = ((♯‘𝑏) + 2))
62	61	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (♯‘𝑐) = ((♯‘𝑏) + 2))
63		simpl3 1191	. . . . . . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . . . . . 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 5096	. . . . . . . . . . . . . . . . . . 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 3619	. . . . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → 𝑏 ∈ ran 𝑆)
68		ffn 6584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆:dom 𝑆⟶𝑊 → 𝑆 Fn dom 𝑆)
69	10, 68	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑆 Fn dom 𝑆
70		fvelrnb 6812	. . . . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . . . 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 19251	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑜 ∈ dom 𝑆 ↔ (𝑜 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑜‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑜))(𝑜‘𝑖) ∈ ran (𝑇‘(𝑜‘(𝑖 − 1)))))
75	74	simp1bi 1143	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑜 ∈ dom 𝑆 → 𝑜 ∈ (Word 𝑊 ∖ {∅}))
76		eldifi 4057	. . . . . . . . . . . . . . . . . . . 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 1190	. . . . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . . . . . 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 6760	. . . . . . . . . . . . . . . . . . . . . 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 5836	. . . . . . . . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . . . . . . 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 19258	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑜 ∈ dom 𝑆 ∧ 𝑐 ∈ ran (𝑇‘(𝑆‘𝑜))) → (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆)
85	73, 83, 84	syl2anc 583	. . . . . . . . . . . . . . . . . . 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 19254	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑜 ∈ Word 𝑊 ∧ 𝑐 ∈ 𝑊 ∧ (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) = 𝑐)
87	77, 78, 85, 86	syl3anc 1369	. . . . . . . . . . . . . . . . . 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 6940	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 Fn dom 𝑆 ∧ (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) ∈ ran 𝑆)
89	69, 85, 88	sylancr 586	. . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . 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 3219	. . . . . . . . . . . . . . 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 3213	. . . . . . . . . . . . . 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	syl5bi 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 3893	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) ↔ (𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)))
96	5	eleq2i 2830	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ 𝐷 ↔ 𝑐 ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)))
97	1, 2, 3, 4, 5, 6	efgs1 19256	. . . . . . . . . . . . . . . . . . . 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 19252	. . . . . . . . . . . . . . . . . 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 14239	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (♯‘⟨“𝑐”⟩) = 1
103	102	oveq1i 7265	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘⟨“𝑐”⟩) − 1) = (1 − 1)
104		1m1e0 11975	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 − 1) = 0
105	103, 104	eqtri 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘⟨“𝑐”⟩) − 1) = 0
106	105	fveq2i 6759	. . . . . . . . . . . . . . . . . 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 14243	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 ∈ 𝑊 → (⟨“𝑐”⟩‘0) = 𝑐)
109	108	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (⟨“𝑐”⟩‘0) = 𝑐)
110	101, 107, 109	3eqtrd 2782	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (𝑆‘⟨“𝑐”⟩) = 𝑐)
111		fnfvelrn 6940	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 Fn dom 𝑆 ∧ ⟨“𝑐”⟩ ∈ dom 𝑆) → (𝑆‘⟨“𝑐”⟩) ∈ ran 𝑆)
112	69, 99, 111	sylancr 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (𝑆‘⟨“𝑐”⟩) ∈ ran 𝑆)
113	110, 112	eqeltrrd 2840	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → 𝑐 ∈ ran 𝑆)
114	113	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝑊 → (¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) → 𝑐 ∈ ran 𝑆))
115	114	3ad2ant2 1132	. . . . . . . . . . . . 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 4004	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑐 ∈ 𝑊 ∣ (♯‘𝑐) = 𝑑} ⊆ ran 𝑆)
118	43, 117	eqsstrid 3965	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) = 𝑑} ⊆ ran 𝑆)
119	41, 118	unssd 4116	. . . . . . . . 9 ⊢ ((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) = 𝑑}) ⊆ ran 𝑆)
120	119	ex 412	. . . . . . . 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 12309	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑎) ∈ ℕ0 ∧ 𝑑 ∈ ℕ0) → ((♯‘𝑎) ≤ 𝑑 ↔ (♯‘𝑎) < (𝑑 + 1)))
123	21, 121, 122	syl2anr 596	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ ℕ0 ∧ 𝑎 ∈ 𝑊) → ((♯‘𝑎) ≤ 𝑑 ↔ (♯‘𝑎) < (𝑑 + 1)))
124	21	nn0red 12224	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑊 → (♯‘𝑎) ∈ ℝ)
125		nn0re 12172	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ0 → 𝑑 ∈ ℝ)
126		leloe 10992	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑎) ∈ ℝ ∧ 𝑑 ∈ ℝ) → ((♯‘𝑎) ≤ 𝑑 ↔ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)))
127	124, 125, 126	syl2anr 596	. . . . . . . . . . . 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 3402	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℕ0 → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} = {𝑎 ∈ 𝑊 ∣ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)})
130		unrab 4236	. . . . . . . . . 10 ⊢ ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) = 𝑑}) = {𝑎 ∈ 𝑊 ∣ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)}
131	129, 130	eqtr4di 2797	. . . . . . . . 9 ⊢ (𝑑 ∈ ℕ0 → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} = ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) = 𝑑}))
132	131	sseq1d 3948	. . . . . . . 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 12345	. . . . . 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 6756	. . . . . . 7 ⊢ (𝑎 = 𝑐 → (♯‘𝑎) = (♯‘𝑐))
137	136	breq1d 5080	. . . . . 6 ⊢ (𝑎 = 𝑐 → ((♯‘𝑎) < ((♯‘𝑐) + 1) ↔ (♯‘𝑐) < ((♯‘𝑐) + 1)))
138		id 22	. . . . . 6 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ 𝑊)
139	17	nn0red 12224	. . . . . . 7 ⊢ (𝑐 ∈ 𝑊 → (♯‘𝑐) ∈ ℝ)
140	139	ltp1d 11835	. . . . . 6 ⊢ (𝑐 ∈ 𝑊 → (♯‘𝑐) < ((♯‘𝑐) + 1))
141	137, 138, 140	elrabd 3619	. . . . 5 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)})
142	135, 141	sseldd 3918	. . . 4 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ ran 𝑆)
143	142	ssriv 3921	. . 3 ⊢ 𝑊 ⊆ ran 𝑆
144	12, 143	eqssi 3933	. 2 ⊢ ran 𝑆 = 𝑊
145		dffo2 6676	. 2 ⊢ (𝑆:dom 𝑆–onto→𝑊 ↔ (𝑆:dom 𝑆⟶𝑊 ∧ ran 𝑆 = 𝑊))
146	10, 144, 145	mpbir2an 707	1 ⊢ 𝑆:dom 𝑆–onto→𝑊

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  {crab 3067   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  {csn 4558  ⟨cop 4564  ⟨cotp 4566  ∪ ciun 4921   class class class wbr 5070   ↦ cmpt 5153   I cid 5479   × cxp 5578  dom cdm 5580  ran crn 5581   Fn wfn 6413  ⟶wf 6414  –onto→wfo 6416  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  1_oc1o 8260  2_oc2o 8261  ℝcr 10801  0cc0 10802  1c1 10803   + caddc 10805   < clt 10940   ≤ cle 10941   − cmin 11135  2c2 11958  ℕ₀cn0 12163  ℝ⁺crp 12659  ...cfz 13168  ..^cfzo 13311  ♯chash 13972  Word cword 14145   ++ cconcat 14201  ⟨“cs1 14228   splice csplice 14390  ⟨“cs2 14482   ~_FG cefg 19227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-rp 12660  df-fz 13169  df-fzo 13312  df-hash 13973  df-word 14146  df-concat 14202  df-s1 14229  df-substr 14282  df-pfx 14312  df-splice 14391  df-s2 14489

This theorem is referenced by:  efgredlemc  19266  efgrelexlemb  19271  efgredeu  19273  efgred2  19274