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Theorem wwlktovfo 14905

Description: Lemma 3 for wrd2f1tovbij 14907. (Contributed by Alexander van der Vekens, 27-Jul-2018.)

**Hypotheses**
Ref	Expression
wwlktovf1o.d	⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}
wwlktovf1o.r	⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}
wwlktovf1o.f	⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1))

**Assertion**
Ref	Expression
wwlktovfo	⊢ (𝑃 ∈ 𝑉 → 𝐹:𝐷–onto→𝑅)

Distinct variable groups: 𝑡,𝐷 𝑃,𝑛,𝑡,𝑤 𝑡,𝑅 𝑛,𝑉,𝑡,𝑤 𝑛,𝑋,𝑤 Allowed substitution hints: 𝐷(𝑤,𝑛) 𝑅(𝑤,𝑛) 𝐹(𝑤,𝑡,𝑛) 𝑋(𝑡)

Proof of Theorem wwlktovfo Dummy variables 𝑝 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlktovf1o.d	. . . 4 ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}
2		wwlktovf1o.r	. . . 4 ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}
3		wwlktovf1o.f	. . . 4 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1))
4	1, 2, 3	wwlktovf 14903	. . 3 ⊢ 𝐹:𝐷⟶𝑅
5	4	a1i 11	. 2 ⊢ (𝑃 ∈ 𝑉 → 𝐹:𝐷⟶𝑅)
6		preq2 4737	. . . . . 6 ⊢ (𝑛 = 𝑝 → {𝑃, 𝑛} = {𝑃, 𝑝})
7	6	eleq1d 2818	. . . . 5 ⊢ (𝑛 = 𝑝 → ({𝑃, 𝑛} ∈ 𝑋 ↔ {𝑃, 𝑝} ∈ 𝑋))
8	7, 2	elrab2 3685	. . . 4 ⊢ (𝑝 ∈ 𝑅 ↔ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))
9		simpl 483	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋) → 𝑝 ∈ 𝑉)
10	9	anim2i 617	. . . . . . . . . 10 ⊢ ((𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋)) → (𝑃 ∈ 𝑉 ∧ 𝑝 ∈ 𝑉))
11		eqidd 2733	. . . . . . . . . 10 ⊢ ((𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋)) → {⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = {⟨0, 𝑃⟩, ⟨1, 𝑝⟩})
12		wrdlen2i 14889	. . . . . . . . . 10 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑝 ∈ 𝑉) → ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = {⟨0, 𝑃⟩, ⟨1, 𝑝⟩} → (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝))))
13	10, 11, 12	sylc 65	. . . . . . . . 9 ⊢ ((𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋)) → (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝)))
14		prex 5431	. . . . . . . . . . 11 ⊢ {⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ V
15	14	a1i 11	. . . . . . . . . 10 ⊢ ((𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋)) → {⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ V)
16		eleq1 2821	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ↔ 𝑢 ∈ Word 𝑉))
17	16	biimpd 228	. . . . . . . . . . . . . . . . . . 19 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 → 𝑢 ∈ Word 𝑉))
18	17	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) → ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 → 𝑢 ∈ Word 𝑉))
19	18	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 → (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) → 𝑢 ∈ Word 𝑉))
20	19	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) → (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) → 𝑢 ∈ Word 𝑉))
21	20	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝)) → (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) → 𝑢 ∈ Word 𝑉))
22	21	impcom 408	. . . . . . . . . . . . . 14 ⊢ ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝))) → 𝑢 ∈ Word 𝑉)
23		fveqeq2 6897	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → ((♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2 ↔ (♯‘𝑢) = 2))
24	23	biimpd 228	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → ((♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2 → (♯‘𝑢) = 2))
25	24	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) → ((♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2 → (♯‘𝑢) = 2))
26	25	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2 → (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) → (♯‘𝑢) = 2))
27	26	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) → (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) → (♯‘𝑢) = 2))
28	27	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝)) → (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) → (♯‘𝑢) = 2))
29	28	impcom 408	. . . . . . . . . . . . . . 15 ⊢ ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝))) → (♯‘𝑢) = 2)
30		fveq1 6887	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = (𝑢‘0))
31	30	eqeq1d 2734	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ↔ (𝑢‘0) = 𝑃))
32	31	biimpd 228	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 → (𝑢‘0) = 𝑃))
33	32	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) → (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 → (𝑢‘0) = 𝑃))
34	33	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 → (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) → (𝑢‘0) = 𝑃))
35	34	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝) → (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) → (𝑢‘0) = 𝑃))
36	35	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝)) → (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) → (𝑢‘0) = 𝑃))
37	36	impcom 408	. . . . . . . . . . . . . . 15 ⊢ ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝))) → (𝑢‘0) = 𝑃)
38		fveq1 6887	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = (𝑢‘1))
39	38	eqeq1d 2734	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝 ↔ (𝑢‘1) = 𝑝))
40	31, 39	anbi12d 631	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝) ↔ ((𝑢‘0) = 𝑃 ∧ (𝑢‘1) = 𝑝)))
41		preq12 4738	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑢‘0) = 𝑃 ∧ (𝑢‘1) = 𝑝) → {(𝑢‘0), (𝑢‘1)} = {𝑃, 𝑝})
42	41	eqcomd 2738	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑢‘0) = 𝑃 ∧ (𝑢‘1) = 𝑝) → {𝑃, 𝑝} = {(𝑢‘0), (𝑢‘1)})
43	42	eleq1d 2818	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑢‘0) = 𝑃 ∧ (𝑢‘1) = 𝑝) → ({𝑃, 𝑝} ∈ 𝑋 ↔ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋))
44	43	biimpd 228	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑢‘0) = 𝑃 ∧ (𝑢‘1) = 𝑝) → ({𝑃, 𝑝} ∈ 𝑋 → {(𝑢‘0), (𝑢‘1)} ∈ 𝑋))
45	40, 44	syl6bi 252	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝) → ({𝑃, 𝑝} ∈ 𝑋 → {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)))
46	45	com12 32	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝) → ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → ({𝑃, 𝑝} ∈ 𝑋 → {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)))
47	46	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝)) → ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → ({𝑃, 𝑝} ∈ 𝑋 → {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)))
48	47	com13 88	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑃, 𝑝} ∈ 𝑋 → ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝)) → {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)))
49	48	ad2antll 727	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋)) → ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝)) → {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)))
50	49	impcom 408	. . . . . . . . . . . . . . . 16 ⊢ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) → ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝)) → {(𝑢‘0), (𝑢‘1)} ∈ 𝑋))
51	50	imp 407	. . . . . . . . . . . . . . 15 ⊢ ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝))) → {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)
52	29, 37, 51	3jca 1128	. . . . . . . . . . . . . 14 ⊢ ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝))) → ((♯‘𝑢) = 2 ∧ (𝑢‘0) = 𝑃 ∧ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋))
53		eqcom 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝 ↔ 𝑝 = ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1))
54	38	eqeq2d 2743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → (𝑝 = ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) ↔ 𝑝 = (𝑢‘1)))
55	54	biimpd 228	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → (𝑝 = ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) → 𝑝 = (𝑢‘1)))
56	53, 55	biimtrid 241	. . . . . . . . . . . . . . . . . . 19 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝 → 𝑝 = (𝑢‘1)))
57	56	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝 → ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → 𝑝 = (𝑢‘1)))
58	57	ad2antll 727	. . . . . . . . . . . . . . . . 17 ⊢ ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝)) → ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → 𝑝 = (𝑢‘1)))
59	58	com12 32	. . . . . . . . . . . . . . . 16 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝)) → 𝑝 = (𝑢‘1)))
60	59	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) → ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝)) → 𝑝 = (𝑢‘1)))
61	60	imp 407	. . . . . . . . . . . . . 14 ⊢ ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝))) → 𝑝 = (𝑢‘1))
62	22, 52, 61	jca31 515	. . . . . . . . . . . . 13 ⊢ ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 ∧ (𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋))) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝))) → ((𝑢 ∈ Word 𝑉 ∧ ((♯‘𝑢) = 2 ∧ (𝑢‘0) = 𝑃 ∧ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)) ∧ 𝑝 = (𝑢‘1)))
63	62	exp31 420	. . . . . . . . . . . 12 ⊢ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} = 𝑢 → ((𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋)) → ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝)) → ((𝑢 ∈ Word 𝑉 ∧ ((♯‘𝑢) = 2 ∧ (𝑢‘0) = 𝑃 ∧ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)) ∧ 𝑝 = (𝑢‘1)))))
64	63	eqcoms 2740	. . . . . . . . . . 11 ⊢ (𝑢 = {⟨0, 𝑃⟩, ⟨1, 𝑝⟩} → ((𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋)) → ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝)) → ((𝑢 ∈ Word 𝑉 ∧ ((♯‘𝑢) = 2 ∧ (𝑢‘0) = 𝑃 ∧ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)) ∧ 𝑝 = (𝑢‘1)))))
65	64	impcom 408	. . . . . . . . . 10 ⊢ (((𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋)) ∧ 𝑢 = {⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) → ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝)) → ((𝑢 ∈ Word 𝑉 ∧ ((♯‘𝑢) = 2 ∧ (𝑢‘0) = 𝑃 ∧ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)) ∧ 𝑝 = (𝑢‘1))))
66	15, 65	spcimedv 3585	. . . . . . . . 9 ⊢ ((𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋)) → ((({⟨0, 𝑃⟩, ⟨1, 𝑝⟩} ∈ Word 𝑉 ∧ (♯‘{⟨0, 𝑃⟩, ⟨1, 𝑝⟩}) = 2) ∧ (({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘0) = 𝑃 ∧ ({⟨0, 𝑃⟩, ⟨1, 𝑝⟩}‘1) = 𝑝)) → ∃𝑢((𝑢 ∈ Word 𝑉 ∧ ((♯‘𝑢) = 2 ∧ (𝑢‘0) = 𝑃 ∧ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)) ∧ 𝑝 = (𝑢‘1))))
67	13, 66	mpd 15	. . . . . . . 8 ⊢ ((𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋)) → ∃𝑢((𝑢 ∈ Word 𝑉 ∧ ((♯‘𝑢) = 2 ∧ (𝑢‘0) = 𝑃 ∧ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)) ∧ 𝑝 = (𝑢‘1)))
68		fveqeq2 6897	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑢 → ((♯‘𝑤) = 2 ↔ (♯‘𝑢) = 2))
69		fveq1 6887	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → (𝑤‘0) = (𝑢‘0))
70	69	eqeq1d 2734	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑢 → ((𝑤‘0) = 𝑃 ↔ (𝑢‘0) = 𝑃))
71		fveq1 6887	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑢 → (𝑤‘1) = (𝑢‘1))
72	69, 71	preq12d 4744	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → {(𝑤‘0), (𝑤‘1)} = {(𝑢‘0), (𝑢‘1)})
73	72	eleq1d 2818	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑢 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋))
74	68, 70, 73	3anbi123d 1436	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑢 → (((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋) ↔ ((♯‘𝑢) = 2 ∧ (𝑢‘0) = 𝑃 ∧ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)))
75	74	elrab 3682	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} ↔ (𝑢 ∈ Word 𝑉 ∧ ((♯‘𝑢) = 2 ∧ (𝑢‘0) = 𝑃 ∧ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)))
76	75	anbi1i 624	. . . . . . . . 9 ⊢ ((𝑢 ∈ {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} ∧ 𝑝 = (𝑢‘1)) ↔ ((𝑢 ∈ Word 𝑉 ∧ ((♯‘𝑢) = 2 ∧ (𝑢‘0) = 𝑃 ∧ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)) ∧ 𝑝 = (𝑢‘1)))
77	76	exbii 1850	. . . . . . . 8 ⊢ (∃𝑢(𝑢 ∈ {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} ∧ 𝑝 = (𝑢‘1)) ↔ ∃𝑢((𝑢 ∈ Word 𝑉 ∧ ((♯‘𝑢) = 2 ∧ (𝑢‘0) = 𝑃 ∧ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)) ∧ 𝑝 = (𝑢‘1)))
78	67, 77	sylibr 233	. . . . . . 7 ⊢ ((𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋)) → ∃𝑢(𝑢 ∈ {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} ∧ 𝑝 = (𝑢‘1)))
79		df-rex 3071	. . . . . . 7 ⊢ (∃𝑢 ∈ {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}𝑝 = (𝑢‘1) ↔ ∃𝑢(𝑢 ∈ {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} ∧ 𝑝 = (𝑢‘1)))
80	78, 79	sylibr 233	. . . . . 6 ⊢ ((𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋)) → ∃𝑢 ∈ {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}𝑝 = (𝑢‘1))
81	1	rexeqi 3324	. . . . . 6 ⊢ (∃𝑢 ∈ 𝐷 𝑝 = (𝑢‘1) ↔ ∃𝑢 ∈ {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}𝑝 = (𝑢‘1))
82	80, 81	sylibr 233	. . . . 5 ⊢ ((𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋)) → ∃𝑢 ∈ 𝐷 𝑝 = (𝑢‘1))
83		fveq1 6887	. . . . . . . 8 ⊢ (𝑡 = 𝑢 → (𝑡‘1) = (𝑢‘1))
84		fvex 6901	. . . . . . . 8 ⊢ (𝑢‘1) ∈ V
85	83, 3, 84	fvmpt 6995	. . . . . . 7 ⊢ (𝑢 ∈ 𝐷 → (𝐹‘𝑢) = (𝑢‘1))
86	85	eqeq2d 2743	. . . . . 6 ⊢ (𝑢 ∈ 𝐷 → (𝑝 = (𝐹‘𝑢) ↔ 𝑝 = (𝑢‘1)))
87	86	rexbiia 3092	. . . . 5 ⊢ (∃𝑢 ∈ 𝐷 𝑝 = (𝐹‘𝑢) ↔ ∃𝑢 ∈ 𝐷 𝑝 = (𝑢‘1))
88	82, 87	sylibr 233	. . . 4 ⊢ ((𝑃 ∈ 𝑉 ∧ (𝑝 ∈ 𝑉 ∧ {𝑃, 𝑝} ∈ 𝑋)) → ∃𝑢 ∈ 𝐷 𝑝 = (𝐹‘𝑢))
89	8, 88	sylan2b 594	. . 3 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑝 ∈ 𝑅) → ∃𝑢 ∈ 𝐷 𝑝 = (𝐹‘𝑢))
90	89	ralrimiva 3146	. 2 ⊢ (𝑃 ∈ 𝑉 → ∀𝑝 ∈ 𝑅 ∃𝑢 ∈ 𝐷 𝑝 = (𝐹‘𝑢))
91		dffo3 7100	. 2 ⊢ (𝐹:𝐷–onto→𝑅 ↔ (𝐹:𝐷⟶𝑅 ∧ ∀𝑝 ∈ 𝑅 ∃𝑢 ∈ 𝐷 𝑝 = (𝐹‘𝑢)))
92	5, 90, 91	sylanbrc 583	1 ⊢ (𝑃 ∈ 𝑉 → 𝐹:𝐷–onto→𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 {crab 3432 Vcvv 3474 {cpr 4629 ⟨cop 4633 ↦ cmpt 5230 ⟶wf 6536 –onto→wfo 6538 ‘cfv 6540 0cc0 11106 1c1 11107 2c2 12263 ♯chash 14286 Word cword 14460

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461

This theorem is referenced by: wwlktovf1o 14906

W3C validator