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Theorem efgsfo 17975

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 17965	. . 3 ⊢ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊
8	7	fdmi 5965	. . . 4 ⊢ dom 𝑆 = {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}
9	8	feq2i 5950	. . 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 220	. 2 ⊢ 𝑆:dom 𝑆⟶𝑊
11		frn 5966	. . . 4 ⊢ (𝑆:dom 𝑆⟶𝑊 → ran 𝑆 ⊆ 𝑊)
12	10, 11	ax-mp 5	. . 3 ⊢ ran 𝑆 ⊆ 𝑊
13		fviss 6166	. . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2_𝑜)) ⊆ Word (𝐼 × 2_𝑜)
14	1, 13	eqsstri 3598	. . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2_𝑜)
15	14	sseli 3564	. . . . . . 7 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ Word (𝐼 × 2_𝑜))
16		lencl 13179	. . . . . . 7 ⊢ (𝑐 ∈ Word (𝐼 × 2_𝑜) → (#‘𝑐) ∈ ℕ₀)
17	15, 16	syl 17	. . . . . 6 ⊢ (𝑐 ∈ 𝑊 → (#‘𝑐) ∈ ℕ₀)
18		peano2nn0 11210	. . . . . 6 ⊢ ((#‘𝑐) ∈ ℕ₀ → ((#‘𝑐) + 1) ∈ ℕ₀)
19	14	sseli 3564	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑊 → 𝑎 ∈ Word (𝐼 × 2_𝑜))
20		lencl 13179	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ Word (𝐼 × 2_𝑜) → (#‘𝑎) ∈ ℕ₀)
21	19, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝑊 → (#‘𝑎) ∈ ℕ₀)
22		nn0nlt0 11196	. . . . . . . . . . . 12 ⊢ ((#‘𝑎) ∈ ℕ₀ → ¬ (#‘𝑎) < 0)
23		breq2 4587	. . . . . . . . . . . . 13 ⊢ (𝑏 = 0 → ((#‘𝑎) < 𝑏 ↔ (#‘𝑎) < 0))
24	23	notbid 307	. . . . . . . . . . . 12 ⊢ (𝑏 = 0 → (¬ (#‘𝑎) < 𝑏 ↔ ¬ (#‘𝑎) < 0))
25	22, 24	syl5ibr 235	. . . . . . . . . . 11 ⊢ (𝑏 = 0 → ((#‘𝑎) ∈ ℕ₀ → ¬ (#‘𝑎) < 𝑏))
26	21, 25	syl5 33	. . . . . . . . . 10 ⊢ (𝑏 = 0 → (𝑎 ∈ 𝑊 → ¬ (#‘𝑎) < 𝑏))
27	26	ralrimiv 2948	. . . . . . . . 9 ⊢ (𝑏 = 0 → ∀𝑎 ∈ 𝑊 ¬ (#‘𝑎) < 𝑏)
28		rabeq0 3911	. . . . . . . . 9 ⊢ ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} = ∅ ↔ ∀𝑎 ∈ 𝑊 ¬ (#‘𝑎) < 𝑏)
29	27, 28	sylibr 223	. . . . . . . 8 ⊢ (𝑏 = 0 → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} = ∅)
30	29	sseq1d 3595	. . . . . . 7 ⊢ (𝑏 = 0 → ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ ∅ ⊆ ran 𝑆))
31		breq2 4587	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → ((#‘𝑎) < 𝑏 ↔ (#‘𝑎) < 𝑑))
32	31	rabbidv 3164	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} = {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑})
33	32	sseq1d 3595	. . . . . . 7 ⊢ (𝑏 = 𝑑 → ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆))
34		breq2 4587	. . . . . . . . 9 ⊢ (𝑏 = (𝑑 + 1) → ((#‘𝑎) < 𝑏 ↔ (#‘𝑎) < (𝑑 + 1)))
35	34	rabbidv 3164	. . . . . . . 8 ⊢ (𝑏 = (𝑑 + 1) → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} = {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < (𝑑 + 1)})
36	35	sseq1d 3595	. . . . . . 7 ⊢ (𝑏 = (𝑑 + 1) → ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆))
37		breq2 4587	. . . . . . . . 9 ⊢ (𝑏 = ((#‘𝑐) + 1) → ((#‘𝑎) < 𝑏 ↔ (#‘𝑎) < ((#‘𝑐) + 1)))
38	37	rabbidv 3164	. . . . . . . 8 ⊢ (𝑏 = ((#‘𝑐) + 1) → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} = {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < ((#‘𝑐) + 1)})
39	38	sseq1d 3595	. . . . . . 7 ⊢ (𝑏 = ((#‘𝑐) + 1) → ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑏} ⊆ ran 𝑆 ↔ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < ((#‘𝑐) + 1)} ⊆ ran 𝑆))
40		0ss 3924	. . . . . . 7 ⊢ ∅ ⊆ ran 𝑆
41		simpr 476	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆)
42		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑐 → (#‘𝑎) = (#‘𝑐))
43	42	eqeq1d 2612	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → ((#‘𝑎) = 𝑑 ↔ (#‘𝑐) = 𝑑))
44	43	cbvrabv 3172	. . . . . . . . . . 11 ⊢ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) = 𝑑} = {𝑐 ∈ 𝑊 ∣ (#‘𝑐) = 𝑑}
45		eliun 4460	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ↔ ∃𝑥 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑥))
46		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑏 → (𝑇‘𝑥) = (𝑇‘𝑏))
47	46	rneqd 5274	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑏 → ran (𝑇‘𝑥) = ran (𝑇‘𝑏))
48	47	eleq2d 2673	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑏 → (𝑐 ∈ ran (𝑇‘𝑥) ↔ 𝑐 ∈ ran (𝑇‘𝑏)))
49	48	cbvrexv 3148	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑥) ↔ ∃𝑏 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑏))
50	45, 49	bitri 263	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ↔ ∃𝑏 ∈ 𝑊 𝑐 ∈ ran (𝑇‘𝑏))
51		simpl1r 1106	. . . . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . . . 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 3566	. . . . . . . . . . . . . . . . . . . . . . 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 13179	. . . . . . . . . . . . . . . . . . . . . . 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 11229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (#‘𝑏) ∈ ℝ)
57		2rp 11713	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ⁺
58		ltaddrp 11743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((#‘𝑏) ∈ ℝ ∧ 2 ∈ ℝ⁺) → (#‘𝑏) < ((#‘𝑏) + 2))
59	56, 57, 58	sylancl 693	. . . . . . . . . . . . . . . . . . . 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 17962	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏)) → (#‘𝑐) = ((#‘𝑏) + 2))
61	60	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (#‘𝑐) = ((#‘𝑏) + 2))
62		simpl3 1059	. . . . . . . . . . . . . . . . . . . . 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 2646	. . . . . . . . . . . . . . . . . . . 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 4609	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → (#‘𝑏) < 𝑑)
65		fveq2 6103	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑏 → (#‘𝑎) = (#‘𝑏))
66	65	breq1d 4593	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑏 → ((#‘𝑎) < 𝑑 ↔ (#‘𝑏) < 𝑑))
67	66	elrab 3331	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ↔ (𝑏 ∈ 𝑊 ∧ (#‘𝑏) < 𝑑))
68	52, 64, 67	sylanbrc 695	. . . . . . . . . . . . . . . . . 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 3569	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) ∧ (𝑏 ∈ 𝑊 ∧ 𝑐 ∈ ran (𝑇‘𝑏))) → 𝑏 ∈ ran 𝑆)
70		ffn 5958	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆:dom 𝑆⟶𝑊 → 𝑆 Fn dom 𝑆)
71	10, 70	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑆 Fn dom 𝑆
72		fvelrnb 6153	. . . . . . . . . . . . . . . . . 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 207	. . . . . . . . . . . . . . . 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 800	. . . . . . . . . . . . . . . . . . . 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 17966	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑜 ∈ dom 𝑆 ↔ (𝑜 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑜‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘𝑜))(𝑜‘𝑖) ∈ ran (𝑇‘(𝑜‘(𝑖 − 1)))))
77	76	simp1bi 1069	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑜 ∈ dom 𝑆 → 𝑜 ∈ (Word 𝑊 ∖ {∅}))
78		eldifi 3694	. . . . . . . . . . . . . . . . . . . 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 1058	. . . . . . . . . . . . . . . . . . 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 799	. . . . . . . . . . . . . . . . . . . . 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 801	. . . . . . . . . . . . . . . . . . . . . . 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 6107	. . . . . . . . . . . . . . . . . . . . . 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 5274	. . . . . . . . . . . . . . . . . . . . 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 2691	. . . . . . . . . . . . . . . . . . . 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 17973	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑜 ∈ dom 𝑆 ∧ 𝑐 ∈ ran (𝑇‘(𝑆‘𝑜))) → (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆)
87	75, 85, 86	syl2anc 691	. . . . . . . . . . . . . . . . . . 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 17969	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑜 ∈ Word 𝑊 ∧ 𝑐 ∈ 𝑊 ∧ (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) = 𝑐)
89	79, 80, 87, 88	syl3anc 1318	. . . . . . . . . . . . . . . . . 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 6264	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 Fn dom 𝑆 ∧ (𝑜 ++ ⟨“𝑐”⟩) ∈ dom 𝑆) → (𝑆‘(𝑜 ++ ⟨“𝑐”⟩)) ∈ ran 𝑆)
91	71, 87, 90	sylancr 694	. . . . . . . . . . . . . . . . . 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 2689	. . . . . . . . . . . . . . . . 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 678	. . . . . . . . . . . . . . . 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 3017	. . . . . . . . . . . . . . 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 3014	. . . . . . . . . . . . . 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 231	. . . . . . . . . . . . 13 ⊢ (((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) → (𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) → 𝑐 ∈ ran 𝑆))
97		eldif 3550	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) ↔ (𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)))
98	5	eleq2i 2680	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ 𝐷 ↔ 𝑐 ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)))
99	1, 2, 3, 4, 5, 6	efgs1 17971	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ 𝐷 → ⟨“𝑐”⟩ ∈ dom 𝑆)
100	98, 99	sylbir 224	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → ⟨“𝑐”⟩ ∈ dom 𝑆)
101	97, 100	sylbir 224	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → ⟨“𝑐”⟩ ∈ dom 𝑆)
102	1, 2, 3, 4, 5, 6	efgsval 17967	. . . . . . . . . . . . . . . . . 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 13238	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (#‘⟨“𝑐”⟩) = 1
105	104	oveq1i 6559	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘⟨“𝑐”⟩) − 1) = (1 − 1)
106		1m1e0 10966	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 − 1) = 0
107	105, 106	eqtri 2632	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘⟨“𝑐”⟩) − 1) = 0
108	107	fveq2i 6106	. . . . . . . . . . . . . . . . . 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 13243	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 ∈ 𝑊 → (⟨“𝑐”⟩‘0) = 𝑐)
111	110	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (⟨“𝑐”⟩‘0) = 𝑐)
112	103, 109, 111	3eqtrd 2648	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (𝑆‘⟨“𝑐”⟩) = 𝑐)
113		fnfvelrn 6264	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 Fn dom 𝑆 ∧ ⟨“𝑐”⟩ ∈ dom 𝑆) → (𝑆‘⟨“𝑐”⟩) ∈ ran 𝑆)
114	71, 101, 113	sylancr 694	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → (𝑆‘⟨“𝑐”⟩) ∈ ran 𝑆)
115	112, 114	eqeltrrd 2689	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ 𝑊 ∧ ¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → 𝑐 ∈ ran 𝑆)
116	115	ex 449	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝑊 → (¬ 𝑐 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) → 𝑐 ∈ ran 𝑆))
117	116	3ad2ant2 1076	. . . . . . . . . . . . 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 169	. . . . . . . . . . . 12 ⊢ (((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) ∧ 𝑐 ∈ 𝑊 ∧ (#‘𝑐) = 𝑑) → 𝑐 ∈ ran 𝑆)
119	118	rabssdv 3645	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑐 ∈ 𝑊 ∣ (#‘𝑐) = 𝑑} ⊆ ran 𝑆)
120	44, 119	syl5eqss 3612	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) = 𝑑} ⊆ ran 𝑆)
121	41, 120	unssd 3751	. . . . . . . . 9 ⊢ ((𝑑 ∈ ℕ₀ ∧ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆) → ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) = 𝑑}) ⊆ ran 𝑆)
122	121	ex 449	. . . . . . . 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 11313	. . . . . . . . . . . . 13 ⊢ (((#‘𝑎) ∈ ℕ₀ ∧ 𝑑 ∈ ℕ₀) → ((#‘𝑎) ≤ 𝑑 ↔ (#‘𝑎) < (𝑑 + 1)))
125	21, 123, 124	syl2anr 494	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ ℕ₀ ∧ 𝑎 ∈ 𝑊) → ((#‘𝑎) ≤ 𝑑 ↔ (#‘𝑎) < (𝑑 + 1)))
126	21	nn0red 11229	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑊 → (#‘𝑎) ∈ ℝ)
127		nn0re 11178	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ ℝ)
128		leloe 10003	. . . . . . . . . . . . 13 ⊢ (((#‘𝑎) ∈ ℝ ∧ 𝑑 ∈ ℝ) → ((#‘𝑎) ≤ 𝑑 ↔ ((#‘𝑎) < 𝑑 ∨ (#‘𝑎) = 𝑑)))
129	126, 127, 128	syl2anr 494	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ ℕ₀ ∧ 𝑎 ∈ 𝑊) → ((#‘𝑎) ≤ 𝑑 ↔ ((#‘𝑎) < 𝑑 ∨ (#‘𝑎) = 𝑑)))
130	125, 129	bitr3d 269	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ ℕ₀ ∧ 𝑎 ∈ 𝑊) → ((#‘𝑎) < (𝑑 + 1) ↔ ((#‘𝑎) < 𝑑 ∨ (#‘𝑎) = 𝑑)))
131	130	rabbidva 3163	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℕ₀ → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < (𝑑 + 1)} = {𝑎 ∈ 𝑊 ∣ ((#‘𝑎) < 𝑑 ∨ (#‘𝑎) = 𝑑)})
132		unrab 3857	. . . . . . . . . 10 ⊢ ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) = 𝑑}) = {𝑎 ∈ 𝑊 ∣ ((#‘𝑎) < 𝑑 ∨ (#‘𝑎) = 𝑑)}
133	131, 132	syl6eqr 2662	. . . . . . . . 9 ⊢ (𝑑 ∈ ℕ₀ → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < (𝑑 + 1)} = ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) = 𝑑}))
134	133	sseq1d 3595	. . . . . . . 8 ⊢ (𝑑 ∈ ℕ₀ → ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆 ↔ ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ∪ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) = 𝑑}) ⊆ ran 𝑆))
135	122, 134	sylibrd 248	. . . . . . 7 ⊢ (𝑑 ∈ ℕ₀ → ({𝑎 ∈ 𝑊 ∣ (#‘𝑎) < 𝑑} ⊆ ran 𝑆 → {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < (𝑑 + 1)} ⊆ ran 𝑆))
136	30, 33, 36, 39, 40, 135	nn0ind 11348	. . . . . 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 11229	. . . . . . 7 ⊢ (𝑐 ∈ 𝑊 → (#‘𝑐) ∈ ℝ)
140	139	ltp1d 10833	. . . . . 6 ⊢ (𝑐 ∈ 𝑊 → (#‘𝑐) < ((#‘𝑐) + 1))
141	42	breq1d 4593	. . . . . . 7 ⊢ (𝑎 = 𝑐 → ((#‘𝑎) < ((#‘𝑐) + 1) ↔ (#‘𝑐) < ((#‘𝑐) + 1)))
142	141	elrab 3331	. . . . . 6 ⊢ (𝑐 ∈ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < ((#‘𝑐) + 1)} ↔ (𝑐 ∈ 𝑊 ∧ (#‘𝑐) < ((#‘𝑐) + 1)))
143	138, 140, 142	sylanbrc 695	. . . . 5 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ {𝑎 ∈ 𝑊 ∣ (#‘𝑎) < ((#‘𝑐) + 1)})
144	137, 143	sseldd 3569	. . . 4 ⊢ (𝑐 ∈ 𝑊 → 𝑐 ∈ ran 𝑆)
145	144	ssriv 3572	. . 3 ⊢ 𝑊 ⊆ ran 𝑆
146	12, 145	eqssi 3584	. 2 ⊢ ran 𝑆 = 𝑊
147		dffo2 6032	. 2 ⊢ (𝑆:dom 𝑆–onto→𝑊 ↔ (𝑆:dom 𝑆⟶𝑊 ∧ ran 𝑆 = 𝑊))
148	10, 146, 147	mpbir2an 957	1 ⊢ 𝑆:dom 𝑆–onto→𝑊

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 {csn 4125 ⟨cop 4131 ⟨cotp 4133 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 I cid 4948 × cxp 5036 dom cdm 5038 ran crn 5039 Fn wfn 5799 ⟶wf 5800 –onto→wfo 5802 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1_𝑜c1o 7440 2_𝑜c2o 7441 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 2c2 10947 ℕ₀cn0 11169 ℝ⁺crp 11708 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 ++ cconcat 13148 ⟨“cs1 13149 splice csplice 13151 ⟨“cs2 13437 ~_FG cefg 17942

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-substr 13158 df-splice 13159 df-s2 13444

This theorem is referenced by: efgredlemc 17981 efgrelexlemb 17986 efgredeu 17988 efgred2 17989

W3C validator