Theorem efgsfo 19714

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 19704	. . 3 ⊢ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊
8	7	fdmi 6679	. . . 4 ⊢ dom 𝑆 = {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}
9	8	feq2i 6660	. . 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 231	. 2 ⊢ 𝑆:dom 𝑆⟶𝑊
11		frn 6675	. . . 4 ⊢ (𝑆:dom 𝑆⟶𝑊 → ran 𝑆 ⊆ 𝑊)
12	10, 11	ax-mp 5	. . 3 ⊢ ran 𝑆 ⊆ 𝑊
13		fviss 6917	. . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
14	1, 13	eqsstri 3968	. . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
15	14	sseli 3917	. . . . . . 7 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ Word (𝐼 × 2o))
16		lencl 14495	. . . . . . 7 ⊢ (𝑐 ∈ Word (𝐼 × 2o) → (♯‘𝑐) ∈ ℕ0)
17	15, 16	syl 17	. . . . . 6 ⊢ (𝑐 ∈ 𝑊 → (♯‘𝑐) ∈ ℕ0)
18		peano2nn0 12477	. . . . . 6 ⊢ ((♯‘𝑐) ∈ ℕ0 → ((♯‘𝑐) + 1) ∈ ℕ0)
19	14	sseli 3917	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑊 → 𝑎 ∈ Word (𝐼 × 2o))
20		lencl 14495	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ Word (𝐼 × 2o) → (♯‘𝑎) ∈ ℕ0)
21	19, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝑊 → (♯‘𝑎) ∈ ℕ0)
22		nn0nlt0 12463	. . . . . . . . . . . 12 ⊢ ((♯‘𝑎) ∈ ℕ0 → ¬ (♯‘𝑎) < 0)
23		breq2 5089	. . . . . . . . . . . . 13 ⊢ (𝑏 = 0 → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < 0))
24	23	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑏 = 0 → (¬ (♯‘𝑎) < 𝑏 ↔ ¬ (♯‘𝑎) < 0))
25	22, 24	imbitrrid 246	. . . . . . . . . . 11 ⊢ (𝑏 = 0 → ((♯‘𝑎) ∈ ℕ0 → ¬ (♯‘𝑎) < 𝑏))
26	21, 25	syl5 34	. . . . . . . . . 10 ⊢ (𝑏 = 0 → (𝑎 ∈ 𝑊 → ¬ (♯‘𝑎) < 𝑏))
27	26	ralrimiv 3128	. . . . . . . . 9 ⊢ (𝑏 = 0 → ∀𝑎 ∈ 𝑊 ¬ (♯‘𝑎) < 𝑏)
28		rabeq0 4328	. . . . . . . . 9 ⊢ ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} = ∅ ↔ ∀𝑎 ∈ 𝑊 ¬ (♯‘𝑎) < 𝑏)
29	27, 28	sylibr 234	. . . . . . . 8 ⊢ (𝑏 = 0 → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} = ∅)
30	29	sseq1d 3953	. . . . . . 7 ⊢ (𝑏 = 0 → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ ∅ ⊆ ran 𝑆))
31		breq2 5089	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < 𝑑))
32	31	rabbidv 3396	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑})
33	32	sseq1d 3953	. . . . . . 7 ⊢ (𝑏 = 𝑑 → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆))
34		breq2 5089	. . . . . . . . 9 ⊢ (𝑏 = (𝑑 + 1) → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < (𝑑 + 1)))
35	34	rabbidv 3396	. . . . . . . 8 ⊢ (𝑏 = (𝑑 + 1) → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)})
36	35	sseq1d 3953	. . . . . . 7 ⊢ (𝑏 = (𝑑 + 1) → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆))
37		breq2 5089	. . . . . . . . 9 ⊢ (𝑏 = ((♯‘𝑐) + 1) → ((♯‘𝑎) < 𝑏 ↔ (♯‘𝑎) < ((♯‘𝑐) + 1)))
38	37	rabbidv 3396	. . . . . . . 8 ⊢ (𝑏 = ((♯‘𝑐) + 1) → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} = {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)})
39	38	sseq1d 3953	. . . . . . 7 ⊢ (𝑏 = ((♯‘𝑐) + 1) → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)} ⊆ ran 𝑆))
40		0ss 4340	. . . . . . 7 ⊢ ∅ ⊆ ran 𝑆
41		simpr 484	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆)
42		fveqeq2 6849	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → ((♯‘𝑎) = 𝑑 ↔ (♯‘𝑐) = 𝑑))
43	42	cbvrabv 3399	. . . . . . . . . . 11 ⊢ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) = 𝑑} = {𝑐 ∈ 𝑊 ∣ (♯‘𝑐) = 𝑑}
44		eliun 4937	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ↔ ∃𝑥 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑥))
45		fveq2 6840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑏 → (𝑇‘𝑥) = (𝑇‘𝑏))
46	45	rneqd 5893	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑏 → ran (𝑇‘𝑥) = ran (𝑇‘𝑏))
47	46	eleq2d 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑏 → (𝑐 ∈ ran (𝑇‘𝑥) ↔ 𝑐 ∈ ran (𝑇‘𝑏)))
48	47	cbvrexvw 3216	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑥) ↔ ∃𝑏 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑏))
49	44, 48	bitri 275	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ↔ ∃𝑏 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑏))
50		simpl1r 1227	. . . . . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑏 → (♯‘𝑎) = (♯‘𝑏))
52	51	breq1d 5095	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑏 → ((♯‘𝑎) < 𝑑 ↔ (♯‘𝑏) < 𝑑))
53		simprl 771	. . . . . . . . . . . . . . . . . . 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 3919	. . . . . . . . . . . . . . . . . . . . . . 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 14495	. . . . . . . . . . . . . . . . . . . . . . 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 12499	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (♯‘𝑏) ∈ ℝ)
58		2rp 12947	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ+
59		ltaddrp 12981	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝑏) ∈ ℝ ∧ 2 ∈ ℝ+) → (♯‘𝑏) < ((♯‘𝑏) + 2))
60	57, 58, 59	sylancl 587	. . . . . . . . . . . . . . . . . . . 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 19701	. . . . . . . . . . . . . . . . . . . . . 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 1195	. . . . . . . . . . . . . . . . . . . . 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 5111	. . . . . . . . . . . . . . . . . . 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 3636	. . . . . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → 𝑏 ∈ ran 𝑆)
68		ffn 6668	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆:dom 𝑆⟶𝑊 → 𝑆 Fn dom 𝑆)
69	10, 68	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑆 Fn dom 𝑆
70		fvelrnb 6900	. . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . . . . 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 19705	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑜 ∈ dom 𝑆 ↔ (𝑜 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑜‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑜))(𝑜‘𝑖) ∈ ran (𝑇‘(𝑜‘(𝑖 − 1)))))
75	74	simp1bi 1146	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑜 ∈ dom 𝑆 → 𝑜 ∈ (Word 𝑊 ∖ {∅}))
76		eldifi 4071	. . . . . . . . . . . . . . . . . . . 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 1194	. . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . . . . . . . . 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 5893	. . . . . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . . . . . 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 19712	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑜 ∈ dom 𝑆 ∧ 𝑐 ∈ ran (𝑇‘(𝑆‘𝑜))) → (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆)
85	73, 83, 84	syl2anc 585	. . . . . . . . . . . . . . . . . . 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 19708	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑜 ∈ Word 𝑊 ∧ 𝑐 ∈ 𝑊 ∧ (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) = 𝑐)
87	77, 78, 85, 86	syl3anc 1374	. . . . . . . . . . . . . . . . . 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 7032	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 Fn dom 𝑆 ∧ (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) ∈ ran 𝑆)
89	69, 85, 88	sylancr 588	. . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . . 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 3144	. . . . . . . . . . . . . . 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 3139	. . . . . . . . . . . . . 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 242	. . . . . . . . . . . . 13 ⊢ (((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (♯‘𝑐) = 𝑑) → (𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) → 𝑐 ∈ ran 𝑆))
95		eldif 3899	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) ↔ (𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)))
96	5	eleq2i 2828	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ 𝐷 ↔ 𝑐 ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)))
97	1, 2, 3, 4, 5, 6	efgs1 19710	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ 𝐷 → ⟨“𝑐”⟩ ∈ dom 𝑆)
98	96, 97	sylbir 235	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → ⟨“𝑐”⟩ ∈ dom 𝑆)
99	95, 98	sylbir 235	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → ⟨“𝑐”⟩ ∈ dom 𝑆)
100	1, 2, 3, 4, 5, 6	efgsval 19706	. . . . . . . . . . . . . . . . . 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 14569	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (♯‘⟨“𝑐”⟩) = 1
103	102	oveq1i 7377	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘⟨“𝑐”⟩) − 1) = (1 − 1)
104		1m1e0 12253	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 − 1) = 0
105	103, 104	eqtri 2759	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘⟨“𝑐”⟩) − 1) = 0
106	105	fveq2i 6843	. . . . . . . . . . . . . . . . . 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 14573	. . . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (𝑆‘⟨“𝑐”⟩) = 𝑐)
111		fnfvelrn 7032	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 Fn dom 𝑆 ∧ ⟨“𝑐”⟩ ∈ dom 𝑆) → (𝑆‘⟨“𝑐”⟩) ∈ ran 𝑆)
112	69, 99, 111	sylancr 588	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (𝑆‘⟨“𝑐”⟩) ∈ ran 𝑆)
113	110, 112	eqeltrrd 2837	. . . . . . . . . . . . . . 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 1135	. . . . . . . . . . . . 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 4014	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑐 ∈ 𝑊 ∣ (♯‘𝑐) = 𝑑} ⊆ ran 𝑆)
118	43, 117	eqsstrid 3960	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ℕ0 ∧ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) = 𝑑} ⊆ ran 𝑆)
119	41, 118	unssd 4132	. . . . . . . . 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 12588	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑎) ∈ ℕ0 ∧ 𝑑 ∈ ℕ0) → ((♯‘𝑎) ≤ 𝑑 ↔ (♯‘𝑎) < (𝑑 + 1)))
123	21, 121, 122	syl2anr 598	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ ℕ0 ∧ 𝑎 ∈ 𝑊) → ((♯‘𝑎) ≤ 𝑑 ↔ (♯‘𝑎) < (𝑑 + 1)))
124	21	nn0red 12499	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑊 → (♯‘𝑎) ∈ ℝ)
125		nn0re 12446	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ0 → 𝑑 ∈ ℝ)
126		leloe 11232	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑎) ∈ ℝ ∧ 𝑑 ∈ ℝ) → ((♯‘𝑎) ≤ 𝑑 ↔ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)))
127	124, 125, 126	syl2anr 598	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ ℕ0 ∧ 𝑎 ∈ 𝑊) → ((♯‘𝑎) ≤ 𝑑 ↔ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)))
128	123, 127	bitr3d 281	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ ℕ0 ∧ 𝑎 ∈ 𝑊) → ((♯‘𝑎) < (𝑑 + 1) ↔ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)))
129	128	rabbidva 3395	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℕ0 → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} = {𝑎 ∈ 𝑊 ∣ ((♯‘𝑎) < 𝑑 ∨ (♯‘𝑎) = 𝑑)})
130		unrab 4255	. . . . . . . . . 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 3953	. . . . . . . 8 ⊢ (𝑑 ∈ ℕ0 → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆 ↔ ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) = 𝑑}) ⊆ ran 𝑆))
133	120, 132	sylibrd 259	. . . . . . 7 ⊢ (𝑑 ∈ ℕ0 → ({𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < 𝑑} ⊆ ran 𝑆 → {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆))
134	30, 33, 36, 39, 40, 133	nn0ind 12624	. . . . . 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 6840	. . . . . . 7 ⊢ (𝑎 = 𝑐 → (♯‘𝑎) = (♯‘𝑐))
137	136	breq1d 5095	. . . . . 6 ⊢ (𝑎 = 𝑐 → ((♯‘𝑎) < ((♯‘𝑐) + 1) ↔ (♯‘𝑐) < ((♯‘𝑐) + 1)))
138		id 22	. . . . . 6 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ 𝑊)
139	17	nn0red 12499	. . . . . . 7 ⊢ (𝑐 ∈ 𝑊 → (♯‘𝑐) ∈ ℝ)
140	139	ltp1d 12086	. . . . . 6 ⊢ (𝑐 ∈ 𝑊 → (♯‘𝑐) < ((♯‘𝑐) + 1))
141	137, 138, 140	elrabd 3636	. . . . 5 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ {𝑎 ∈ 𝑊 ∣ (♯‘𝑎) < ((♯‘𝑐) + 1)})
142	135, 141	sseldd 3922	. . . 4 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ ran 𝑆)
143	142	ssriv 3925	. . 3 ⊢ 𝑊 ⊆ ran 𝑆
144	12, 143	eqssi 3938	. 2 ⊢ ran 𝑆 = 𝑊
145		dffo2 6756	. 2 ⊢ (𝑆:dom 𝑆–onto→𝑊 ↔ (𝑆:dom 𝑆⟶𝑊 ∧ ran 𝑆 = 𝑊))
146	10, 144, 145	mpbir2an 712	1 ⊢ 𝑆:dom 𝑆–onto→𝑊

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  {crab 3389   ∖ cdif 3886   ∪ cun 3887   ⊆ wss 3889  ∅c0 4273  {csn 4567  ⟨cop 4573  ⟨cotp 4575  ∪ ciun 4933   class class class wbr 5085   ↦ cmpt 5166   I cid 5525   × cxp 5629  dom cdm 5631  ran crn 5632   Fn wfn 6493  ⟶wf 6494  –onto→wfo 6496  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  1_oc1o 8398  2_oc2o 8399  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   < clt 11179   ≤ cle 11180   − cmin 11377  2c2 12236  ℕ₀cn0 12437  ℝ⁺crp 12942  ...cfz 13461  ..^cfzo 13608  ♯chash 14292  Word cword 14475   ++ cconcat 14532  ⟨“cs1 14558   splice csplice 14711  ⟨“cs2 14803   ~_FG cefg 19681

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-rp 12943  df-fz 13462  df-fzo 13609  df-hash 14293  df-word 14476  df-concat 14533  df-s1 14559  df-substr 14604  df-pfx 14634  df-splice 14712  df-s2 14810

This theorem is referenced by:  efgredlemc  19720  efgrelexlemb  19725  efgredeu  19727  efgred2  19728