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Theorem efgsfo 18152

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 indexes somewhere in the sequence). (Contributed by Mario Carneiro, 27-Sep-2015.)

**Hypotheses**
Ref	Expression
efgval.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
efgval.r	⊢ ∼ = ( ~_FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (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 × 2_𝑜))
2		efgval.r	. . . 4 ⊢ ∼ = ( ~_FG ‘𝐼)
3		efgval2.m	. . . 4 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
4		efgval2.t	. . . 4 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (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 18142	. . 3 ⊢ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊
8	7	fdmi 6052	. . . 4 ⊢ dom 𝑆 = {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}
9	8	feq2i 6037	. . 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 221	. 2 ⊢ 𝑆:dom 𝑆⟶𝑊
11		frn 6053	. . . 4 ⊢ (𝑆:dom 𝑆⟶𝑊 → ran 𝑆 ⊆ 𝑊)
12	10, 11	ax-mp 5	. . 3 ⊢ ran 𝑆 ⊆ 𝑊
13		fviss 6256	. . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2_𝑜)) ⊆ Word (𝐼 × 2_𝑜)
14	1, 13	eqsstri 3635	. . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2_𝑜)
15	14	sseli 3599	. . . . . . 7 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ Word (𝐼 × 2_𝑜))
16		lencl 13324	. . . . . . 7 ⊢ (𝑐 ∈ Word (𝐼 × 2_𝑜) → (#‘𝑐) ∈ ℕ₀)
17	15, 16	syl 17	. . . . . 6 ⊢ (𝑐 ∈ 𝑊 → (#‘𝑐) ∈ ℕ₀)
18		peano2nn0 11333	. . . . . 6 ⊢ ((#‘𝑐) ∈ ℕ₀ → ((#‘𝑐) + 1) ∈ ℕ₀)
19	14	sseli 3599	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑊 → 𝑎 ∈ Word (𝐼 × 2_𝑜))
20		lencl 13324	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ Word (𝐼 × 2_𝑜) → (#‘𝑎) ∈ ℕ₀)
21	19, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝑊 → (#‘𝑎) ∈ ℕ₀)
22		nn0nlt0 11319	. . . . . . . . . . . 12 ⊢ ((#‘𝑎) ∈ ℕ₀ → ¬ (#‘𝑎) < 0)
23		breq2 4657	. . . . . . . . . . . . 13 ⊢ (𝑏 = 0 → ((#‘𝑎) < 𝑏 ↔ (#‘𝑎) < 0))
24	23	notbid 308	. . . . . . . . . . . 12 ⊢ (𝑏 = 0 → (¬ (#‘𝑎) < 𝑏 ↔ ¬ (#‘𝑎) < 0))
25	22, 24	syl5ibr 236	. . . . . . . . . . 11 ⊢ (𝑏 = 0 → ((#‘𝑎) ∈ ℕ₀ → ¬ (#‘𝑎) < 𝑏))
26	21, 25	syl5 34	. . . . . . . . . 10 ⊢ (𝑏 = 0 → (𝑎 ∈ 𝑊 → ¬ (#‘𝑎) < 𝑏))
27	26	ralrimiv 2965	. . . . . . . . 9 ⊢ (𝑏 = 0 → ∀𝑎 ∈ 𝑊 ¬ (#‘𝑎) < 𝑏)
28		rabeq0 3957	. . . . . . . . 9 ⊢ ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} = ∅ ↔ ∀𝑎 ∈ 𝑊 ¬ (#‘𝑎) < 𝑏)
29	27, 28	sylibr 224	. . . . . . . 8 ⊢ (𝑏 = 0 → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} = ∅)
30	29	sseq1d 3632	. . . . . . 7 ⊢ (𝑏 = 0 → ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ ∅ ⊆ ran 𝑆))
31		breq2 4657	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → ((#‘𝑎) < 𝑏 ↔ (#‘𝑎) < 𝑑))
32	31	rabbidv 3189	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} = {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑})
33	32	sseq1d 3632	. . . . . . 7 ⊢ (𝑏 = 𝑑 → ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆))
34		breq2 4657	. . . . . . . . 9 ⊢ (𝑏 = (𝑑 + 1) → ((#‘𝑎) < 𝑏 ↔ (#‘𝑎) < (𝑑 + 1)))
35	34	rabbidv 3189	. . . . . . . 8 ⊢ (𝑏 = (𝑑 + 1) → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} = {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < (𝑑 + 1)})
36	35	sseq1d 3632	. . . . . . 7 ⊢ (𝑏 = (𝑑 + 1) → ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆))
37		breq2 4657	. . . . . . . . 9 ⊢ (𝑏 = ((#‘𝑐) + 1) → ((#‘𝑎) < 𝑏 ↔ (#‘𝑎) < ((#‘𝑐) + 1)))
38	37	rabbidv 3189	. . . . . . . 8 ⊢ (𝑏 = ((#‘𝑐) + 1) → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} = {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < ((#‘𝑐) + 1)})
39	38	sseq1d 3632	. . . . . . 7 ⊢ (𝑏 = ((#‘𝑐) + 1) → ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < ((#‘𝑐) + 1)} ⊆ ran 𝑆))
40		0ss 3972	. . . . . . 7 ⊢ ∅ ⊆ ran 𝑆
41		simpr 477	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆)
42		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑐 → (#‘𝑎) = (#‘𝑐))
43	42	eqeq1d 2624	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → ((#‘𝑎) = 𝑑 ↔ (#‘𝑐) = 𝑑))
44	43	cbvrabv 3199	. . . . . . . . . . 11 ⊢ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) = 𝑑} = {𝑐 ∈ 𝑊 ∣ (#‘𝑐) = 𝑑}
45		eliun 4524	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ↔ ∃𝑥 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑥))
46		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑏 → (𝑇‘𝑥) = (𝑇‘𝑏))
47	46	rneqd 5353	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑏 → ran (𝑇‘𝑥) = ran (𝑇‘𝑏))
48	47	eleq2d 2687	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑏 → (𝑐 ∈ ran (𝑇‘𝑥) ↔ 𝑐 ∈ ran (𝑇‘𝑏)))
49	48	cbvrexv 3172	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑥) ↔ ∃𝑏 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑏))
50	45, 49	bitri 264	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ↔ ∃𝑏 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑏))
51		simpl1r 1113	. . . . . . . . . . . . . . . . . 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)
52		simprl 794	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → 𝑏 ∈ 𝑊)
53	14, 52	sseldi 3601	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → 𝑏 ∈ Word (𝐼 × 2_𝑜))
54		lencl 13324	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 ∈ Word (𝐼 × 2_𝑜) → (#‘𝑏) ∈ ℕ₀)
55	53, 54	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (#‘𝑏) ∈ ℕ₀)
56	55	nn0red 11352	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (#‘𝑏) ∈ ℝ)
57		2rp 11837	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ⁺
58		ltaddrp 11867	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((#‘𝑏) ∈ ℝ ∧ 2 ∈ ℝ⁺) → (#‘𝑏) < ((#‘𝑏) + 2))
59	56, 57, 58	sylancl 694	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (#‘𝑏) < ((#‘𝑏) + 2))
60	1, 2, 3, 4	efgtlen 18139	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏)) → (#‘𝑐) = ((#‘𝑏) + 2))
61	60	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (#‘𝑐) = ((#‘𝑏) + 2))
62		simpl3 1066	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (#‘𝑐) = 𝑑)
63	61, 62	eqtr3d 2658	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → ((#‘𝑏) + 2) = 𝑑)
64	59, 63	breqtrd 4679	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (#‘𝑏) < 𝑑)
65		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑏 → (#‘𝑎) = (#‘𝑏))
66	65	breq1d 4663	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑏 → ((#‘𝑎) < 𝑑 ↔ (#‘𝑏) < 𝑑))
67	66	elrab 3363	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ↔ (𝑏 ∈ 𝑊 ∧ (#‘𝑏) < 𝑑))
68	52, 64, 67	sylanbrc 698	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → 𝑏 ∈ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑})
69	51, 68	sseldd 3604	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → 𝑏 ∈ ran 𝑆)
70		ffn 6045	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆:dom 𝑆⟶𝑊 → 𝑆 Fn dom 𝑆)
71	10, 70	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑆 Fn dom 𝑆
72		fvelrnb 6243	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 Fn dom 𝑆 → (𝑏 ∈ ran 𝑆 ↔ ∃𝑜 ∈ dom 𝑆(𝑆‘𝑜) = 𝑏))
73	71, 72	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ ran 𝑆 ↔ ∃𝑜 ∈ dom 𝑆(𝑆‘𝑜) = 𝑏)
74	69, 73	sylib 208	. . . . . . . . . . . . . . . 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)
75		simprrl 804	. . . . . . . . . . . . . . . . . . . 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)
76	1, 2, 3, 4, 5, 6	efgsdm 18143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑜 ∈ dom 𝑆 ↔ (𝑜 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑜‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘𝑜))(𝑜‘𝑖) ∈ ran (𝑇‘(𝑜‘(𝑖 − 1)))))
77	76	simp1bi 1076	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑜 ∈ dom 𝑆 → 𝑜 ∈ (Word 𝑊 ∖ {∅}))
78		eldifi 3732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑜 ∈ (Word 𝑊 ∖ {∅}) → 𝑜 ∈ Word 𝑊)
79	75, 77, 78	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)
80		simpl2 1065	. . . . . . . . . . . . . . . . . . 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)
81		simprlr 803	. . . . . . . . . . . . . . . . . . . . 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))
82		simprrr 805	. . . . . . . . . . . . . . . . . . . . . . 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)
83	82	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . 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))
84	83	rneqd 5353	. . . . . . . . . . . . . . . . . . . . 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))
85	81, 84	eleqtrrd 2704	. . . . . . . . . . . . . . . . . . . 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)))
86	1, 2, 3, 4, 5, 6	efgsp1 18150	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑜 ∈ dom 𝑆 ∧ 𝑐 ∈ ran (𝑇‘(𝑆‘𝑜))) → (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆)
87	75, 85, 86	syl2anc 693	. . . . . . . . . . . . . . . . . . 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)
88	1, 2, 3, 4, 5, 6	efgsval2 18146	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑜 ∈ Word 𝑊 ∧ 𝑐 ∈ 𝑊 ∧ (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) = 𝑐)
89	79, 80, 87, 88	syl3anc 1326	. . . . . . . . . . . . . . . . . 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)
90		fnfvelrn 6356	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 Fn dom 𝑆 ∧ (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) ∈ ran 𝑆)
91	71, 87, 90	sylancr 695	. . . . . . . . . . . . . . . . . 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)
92	89, 91	eqeltrrd 2702	. . . . . . . . . . . . . . . . 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)
93	92	anassrs 680	. . . . . . . . . . . . . . . 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)
94	74, 93	rexlimddv 3035	. . . . . . . . . . . . . . 15 ⊢ ((((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → 𝑐 ∈ ran 𝑆)
95	94	rexlimdvaa 3032	. . . . . . . . . . . . . 14 ⊢ (((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) → (∃𝑏 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑏) → 𝑐 ∈ ran 𝑆))
96	50, 95	syl5bi 232	. . . . . . . . . . . . 13 ⊢ (((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) → (𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) → 𝑐 ∈ ran 𝑆))
97		eldif 3584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) ↔ (𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)))
98	5	eleq2i 2693	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ 𝐷 ↔ 𝑐 ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)))
99	1, 2, 3, 4, 5, 6	efgs1 18148	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ 𝐷 → ⟨“𝑐”⟩ ∈ dom 𝑆)
100	98, 99	sylbir 225	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → ⟨“𝑐”⟩ ∈ dom 𝑆)
101	97, 100	sylbir 225	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → ⟨“𝑐”⟩ ∈ dom 𝑆)
102	1, 2, 3, 4, 5, 6	efgsval 18144	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨“𝑐”⟩ ∈ dom 𝑆 → (𝑆‘⟨“𝑐”⟩) = (⟨“𝑐”⟩‘((#‘⟨“𝑐”⟩) − 1)))
103	101, 102	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (𝑆‘⟨“𝑐”⟩) = (⟨“𝑐”⟩‘((#‘⟨“𝑐”⟩) − 1)))
104		s1len 13385	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (#‘⟨“𝑐”⟩) = 1
105	104	oveq1i 6660	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘⟨“𝑐”⟩) − 1) = (1 − 1)
106		1m1e0 11089	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 − 1) = 0
107	105, 106	eqtri 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘⟨“𝑐”⟩) − 1) = 0
108	107	fveq2i 6194	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨“𝑐”⟩‘((#‘⟨“𝑐”⟩) − 1)) = (⟨“𝑐”⟩‘0)
109	108	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (⟨“𝑐”⟩‘((#‘⟨“𝑐”⟩) − 1)) = (⟨“𝑐”⟩‘0))
110		s1fv 13390	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 ∈ 𝑊 → (⟨“𝑐”⟩‘0) = 𝑐)
111	110	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (⟨“𝑐”⟩‘0) = 𝑐)
112	103, 109, 111	3eqtrd 2660	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (𝑆‘⟨“𝑐”⟩) = 𝑐)
113		fnfvelrn 6356	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 Fn dom 𝑆 ∧ ⟨“𝑐”⟩ ∈ dom 𝑆) → (𝑆‘⟨“𝑐”⟩) ∈ ran 𝑆)
114	71, 101, 113	sylancr 695	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (𝑆‘⟨“𝑐”⟩) ∈ ran 𝑆)
115	112, 114	eqeltrrd 2702	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → 𝑐 ∈ ran 𝑆)
116	115	ex 450	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝑊 → (¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) → 𝑐 ∈ ran 𝑆))
117	116	3ad2ant2 1083	. . . . . . . . . . . . 13 ⊢ (((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) → (¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) → 𝑐 ∈ ran 𝑆))
118	96, 117	pm2.61d 170	. . . . . . . . . . . 12 ⊢ (((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) → 𝑐 ∈ ran 𝑆)
119	118	rabssdv 3682	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑐 ∈ 𝑊 ∣ (#‘𝑐) = 𝑑} ⊆ ran 𝑆)
120	44, 119	syl5eqss 3649	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) = 𝑑} ⊆ ran 𝑆)
121	41, 120	unssd 3789	. . . . . . . . 9 ⊢ ((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) → ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) = 𝑑}) ⊆ ran 𝑆)
122	121	ex 450	. . . . . . . 8 ⊢ (𝑑 ∈ ℕ₀ → ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆 → ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) = 𝑑}) ⊆ ran 𝑆))
123		id 22	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ ℕ₀)
124		nn0leltp1 11436	. . . . . . . . . . . . 13 ⊢ (((#‘𝑎) ∈ ℕ₀ ∧ 𝑑 ∈ ℕ₀) → ((#‘𝑎) ≤ 𝑑 ↔ (#‘𝑎) < (𝑑 + 1)))
125	21, 123, 124	syl2anr 495	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ ℕ₀ ∧ 𝑎 ∈ 𝑊) → ((#‘𝑎) ≤ 𝑑 ↔ (#‘𝑎) < (𝑑 + 1)))
126	21	nn0red 11352	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑊 → (#‘𝑎) ∈ ℝ)
127		nn0re 11301	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ ℝ)
128		leloe 10124	. . . . . . . . . . . . 13 ⊢ (((#‘𝑎) ∈ ℝ ∧ 𝑑 ∈ ℝ) → ((#‘𝑎) ≤ 𝑑 ↔ ((#‘𝑎) < 𝑑 ∨ (#‘𝑎) = 𝑑)))
129	126, 127, 128	syl2anr 495	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ ℕ₀ ∧ 𝑎 ∈ 𝑊) → ((#‘𝑎) ≤ 𝑑 ↔ ((#‘𝑎) < 𝑑 ∨ (#‘𝑎) = 𝑑)))
130	125, 129	bitr3d 270	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ ℕ₀ ∧ 𝑎 ∈ 𝑊) → ((#‘𝑎) < (𝑑 + 1) ↔ ((#‘𝑎) < 𝑑 ∨ (#‘𝑎) = 𝑑)))
131	130	rabbidva 3188	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℕ₀ → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < (𝑑 + 1)} = {𝑎 ∈ 𝑊 ∣ ((#‘𝑎) < 𝑑 ∨ (#‘𝑎) = 𝑑)})
132		unrab 3898	. . . . . . . . . 10 ⊢ ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) = 𝑑}) = {𝑎 ∈ 𝑊 ∣ ((#‘𝑎) < 𝑑 ∨ (#‘𝑎) = 𝑑)}
133	131, 132	syl6eqr 2674	. . . . . . . . 9 ⊢ (𝑑 ∈ ℕ₀ → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < (𝑑 + 1)} = ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) = 𝑑}))
134	133	sseq1d 3632	. . . . . . . 8 ⊢ (𝑑 ∈ ℕ₀ → ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆 ↔ ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) = 𝑑}) ⊆ ran 𝑆))
135	122, 134	sylibrd 249	. . . . . . 7 ⊢ (𝑑 ∈ ℕ₀ → ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆 → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆))
136	30, 33, 36, 39, 40, 135	nn0ind 11472	. . . . . 6 ⊢ (((#‘𝑐) + 1) ∈ ℕ₀ → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < ((#‘𝑐) + 1)} ⊆ ran 𝑆)
137	17, 18, 136	3syl 18	. . . . 5 ⊢ (𝑐 ∈ 𝑊 → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < ((#‘𝑐) + 1)} ⊆ ran 𝑆)
138		id 22	. . . . . 6 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ 𝑊)
139	17	nn0red 11352	. . . . . . 7 ⊢ (𝑐 ∈ 𝑊 → (#‘𝑐) ∈ ℝ)
140	139	ltp1d 10954	. . . . . 6 ⊢ (𝑐 ∈ 𝑊 → (#‘𝑐) < ((#‘𝑐) + 1))
141	42	breq1d 4663	. . . . . . 7 ⊢ (𝑎 = 𝑐 → ((#‘𝑎) < ((#‘𝑐) + 1) ↔ (#‘𝑐) < ((#‘𝑐) + 1)))
142	141	elrab 3363	. . . . . 6 ⊢ (𝑐 ∈ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < ((#‘𝑐) + 1)} ↔ (𝑐 ∈ 𝑊 ∧ (#‘𝑐) < ((#‘𝑐) + 1)))
143	138, 140, 142	sylanbrc 698	. . . . 5 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < ((#‘𝑐) + 1)})
144	137, 143	sseldd 3604	. . . 4 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ ran 𝑆)
145	144	ssriv 3607	. . 3 ⊢ 𝑊 ⊆ ran 𝑆
146	12, 145	eqssi 3619	. 2 ⊢ ran 𝑆 = 𝑊
147		dffo2 6119	. 2 ⊢ (𝑆:dom 𝑆–onto→𝑊 ↔ (𝑆:dom 𝑆⟶𝑊 ∧ ran 𝑆 = 𝑊))
148	10, 146, 147	mpbir2an 955	1 ⊢ 𝑆:dom 𝑆–onto→𝑊

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 {crab 2916 ∖ cdif 3571 ∪ cun 3572 ⊆ wss 3574 ∅c0 3915 {csn 4177 ⟨cop 4183 ⟨cotp 4185 ∪ ciun 4520 class class class wbr 4653 ↦ cmpt 4729 I cid 5023 × cxp 5112 dom cdm 5114 ran crn 5115 Fn wfn 5883 ⟶wf 5884 –onto→wfo 5886 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1_𝑜c1o 7553 2_𝑜c2o 7554 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 2c2 11070 ℕ₀cn0 11292 ℝ⁺crp 11832 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 ++ cconcat 13293 ⟨“cs1 13294 splice csplice 13296 ⟨“cs2 13586 ~_FG cefg 18119

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-s2 13593

This theorem is referenced by: efgredlemc 18158 efgrelexlemb 18163 efgredeu 18165 efgred2 18166

W3C validator