Theorem wlkwwlksurOLD 27156

Description: Obsolete as of 2-Aug-2022. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 16-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
wlkwwlkbijOLD.t	⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}
wlkwwlkbijOLD.w	⊢ 𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
wlkwwlkbijOLD.f	⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡))

Assertion

Ref	Expression
wlkwwlksurOLD	⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–onto→𝑊)

Distinct variable groups:   𝐺,𝑝,𝑡,𝑤   𝑁,𝑝,𝑡,𝑤   𝑃,𝑝,𝑡,𝑤   𝑡,𝑇,𝑤   𝑡,𝑉   𝑡,𝑊   𝑤,𝐹   𝑤,𝑉   𝐹,𝑝   𝑇,𝑝   𝑊,𝑝

Allowed substitution hints:   𝐹(𝑡)   𝑉(𝑝)   𝑊(𝑤)

Proof of Theorem wlkwwlksurOLD

Dummy variables 𝑢 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uspgrupgr 26412	. . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
2		wlkwwlkbijOLD.t	. . . 4 ⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}
3		wlkwwlkbijOLD.w	. . . 4 ⊢ 𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
4		wlkwwlkbijOLD.f	. . . 4 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡))
5	2, 3, 4	wlkwwlkfunOLD 27154	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)
6	1, 5	syl3an1 1203	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)
7		fveq1 6410	. . . . . . 7 ⊢ (𝑤 = 𝑝 → (𝑤‘0) = (𝑝‘0))
8	7	eqeq1d 2801	. . . . . 6 ⊢ (𝑤 = 𝑝 → ((𝑤‘0) = 𝑃 ↔ (𝑝‘0) = 𝑃))
9	8, 3	elrab2 3560	. . . . 5 ⊢ (𝑝 ∈ 𝑊 ↔ (𝑝 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑝‘0) = 𝑃))
10		wlklnwwlkn 27142	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) ↔ 𝑝 ∈ (𝑁 WWalksN 𝐺)))
11		df-br 4844	. . . . . . . . . . . . 13 ⊢ (𝑓(Walks‘𝐺)𝑝 ↔ ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺))
12		vex 3388	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
13		vex 3388	. . . . . . . . . . . . . . . . 17 ⊢ 𝑝 ∈ V
14	12, 13	op1st 7409	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘⟨𝑓, 𝑝⟩) = 𝑓
15	14	eqcomi 2808	. . . . . . . . . . . . . . 15 ⊢ 𝑓 = (1st ‘⟨𝑓, 𝑝⟩)
16	15	fveq2i 6414	. . . . . . . . . . . . . 14 ⊢ (♯‘𝑓) = (♯‘(1st ‘⟨𝑓, 𝑝⟩))
17	16	eqeq1i 2804	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑓) = 𝑁 ↔ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁)
18	12, 13	op2nd 7410	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘⟨𝑓, 𝑝⟩) = 𝑝
19	18	eqcomi 2808	. . . . . . . . . . . . . . . 16 ⊢ 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩)
20	19	fveq1i 6412	. . . . . . . . . . . . . . 15 ⊢ (𝑝‘0) = ((2nd ‘⟨𝑓, 𝑝⟩)‘0)
21	20	eqeq1i 2804	. . . . . . . . . . . . . 14 ⊢ ((𝑝‘0) = 𝑃 ↔ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃)
22		opex 5123	. . . . . . . . . . . . . . . 16 ⊢ ⟨𝑓, 𝑝⟩ ∈ V
23	22	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ⟨𝑓, 𝑝⟩ ∈ V)
24		simpll 784	. . . . . . . . . . . . . . . . . 18 ⊢ (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃) → ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺))
25		simpr 478	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁)
26	25	anim1i 609	. . . . . . . . . . . . . . . . . 18 ⊢ (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃) → ((♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃))
27	19	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃) → 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩))
28	24, 26, 27	jca31 511	. . . . . . . . . . . . . . . . 17 ⊢ (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃) → ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩)))
29		eleq1 2866	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → (𝑢 ∈ (Walks‘𝐺) ↔ ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺)))
30		fveq2 6411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → (1st ‘𝑢) = (1st ‘⟨𝑓, 𝑝⟩))
31	30	fveq2d 6415	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → (♯‘(1st ‘𝑢)) = (♯‘(1st ‘⟨𝑓, 𝑝⟩)))
32	31	eqeq1d 2801	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → ((♯‘(1st ‘𝑢)) = 𝑁 ↔ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁))
33		fveq2 6411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → (2nd ‘𝑢) = (2nd ‘⟨𝑓, 𝑝⟩))
34	33	fveq1d 6413	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → ((2nd ‘𝑢)‘0) = ((2nd ‘⟨𝑓, 𝑝⟩)‘0))
35	34	eqeq1d 2801	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → (((2nd ‘𝑢)‘0) = 𝑃 ↔ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃))
36	32, 35	anbi12d 625	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → (((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃) ↔ ((♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃)))
37	29, 36	anbi12d 625	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → ((𝑢 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)) ↔ (⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃))))
38	33	eqeq2d 2809	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → (𝑝 = (2nd ‘𝑢) ↔ 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩)))
39	37, 38	anbi12d 625	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → (((𝑢 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢)) ↔ ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘⟨𝑓, 𝑝⟩))))
40	28, 39	syl5ibrcom 239	. . . . . . . . . . . . . . . 16 ⊢ (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ ((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃) → (𝑢 = ⟨𝑓, 𝑝⟩ → ((𝑢 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢))))
41	40	impancom 444	. . . . . . . . . . . . . . 15 ⊢ (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → (((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃 → ((𝑢 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢))))
42	23, 41	spcimedv 3480	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → (((2nd ‘⟨𝑓, 𝑝⟩)‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢))))
43	21, 42	syl5bi 234	. . . . . . . . . . . . 13 ⊢ ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢))))
44	11, 17, 43	syl2anb 592	. . . . . . . . . . . 12 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢))))
45	44	exlimiv 2026	. . . . . . . . . . 11 ⊢ (∃𝑓(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢))))
46	10, 45	syl6bir 246	. . . . . . . . . 10 ⊢ (𝐺 ∈ USPGraph → (𝑝 ∈ (𝑁 WWalksN 𝐺) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢)))))
47	46	imp32 410	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑝 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢)))
48		fveq2 6411	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑢 → (1st ‘𝑝) = (1st ‘𝑢))
49	48	fveq2d 6415	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑢 → (♯‘(1st ‘𝑝)) = (♯‘(1st ‘𝑢)))
50	49	eqeq1d 2801	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑢 → ((♯‘(1st ‘𝑝)) = 𝑁 ↔ (♯‘(1st ‘𝑢)) = 𝑁))
51		fveq2 6411	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑢 → (2nd ‘𝑝) = (2nd ‘𝑢))
52	51	fveq1d 6413	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑢 → ((2nd ‘𝑝)‘0) = ((2nd ‘𝑢)‘0))
53	52	eqeq1d 2801	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑢 → (((2nd ‘𝑝)‘0) = 𝑃 ↔ ((2nd ‘𝑢)‘0) = 𝑃))
54	50, 53	anbi12d 625	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑢 → (((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃) ↔ ((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)))
55	54	elrab 3556	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)} ↔ (𝑢 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)))
56	55	anbi1i 618	. . . . . . . . . 10 ⊢ ((𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd ‘𝑢)) ↔ ((𝑢 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢)))
57	56	exbii 1944	. . . . . . . . 9 ⊢ (∃𝑢(𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd ‘𝑢)) ↔ ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢)))
58	47, 57	sylibr 226	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑝 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢(𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd ‘𝑢)))
59		df-rex 3095	. . . . . . . 8 ⊢ (∃𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}𝑝 = (2nd ‘𝑢) ↔ ∃𝑢(𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd ‘𝑢)))
60	58, 59	sylibr 226	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑝 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}𝑝 = (2nd ‘𝑢))
61	2	rexeqi 3326	. . . . . . 7 ⊢ (∃𝑢 ∈ 𝑇 𝑝 = (2nd ‘𝑢) ↔ ∃𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}𝑝 = (2nd ‘𝑢))
62	60, 61	sylibr 226	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑝 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢 ∈ 𝑇 𝑝 = (2nd ‘𝑢))
63		fveq2 6411	. . . . . . . . 9 ⊢ (𝑡 = 𝑢 → (2nd ‘𝑡) = (2nd ‘𝑢))
64		fvex 6424	. . . . . . . . 9 ⊢ (2nd ‘𝑢) ∈ V
65	63, 4, 64	fvmpt 6507	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑇 → (𝐹‘𝑢) = (2nd ‘𝑢))
66	65	eqeq2d 2809	. . . . . . 7 ⊢ (𝑢 ∈ 𝑇 → (𝑝 = (𝐹‘𝑢) ↔ 𝑝 = (2nd ‘𝑢)))
67	66	rexbiia 3221	. . . . . 6 ⊢ (∃𝑢 ∈ 𝑇 𝑝 = (𝐹‘𝑢) ↔ ∃𝑢 ∈ 𝑇 𝑝 = (2nd ‘𝑢))
68	62, 67	sylibr 226	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑝 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢 ∈ 𝑇 𝑝 = (𝐹‘𝑢))
69	9, 68	sylan2b 588	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑝 ∈ 𝑊) → ∃𝑢 ∈ 𝑇 𝑝 = (𝐹‘𝑢))
70	69	ralrimiva 3147	. . 3 ⊢ (𝐺 ∈ USPGraph → ∀𝑝 ∈ 𝑊 ∃𝑢 ∈ 𝑇 𝑝 = (𝐹‘𝑢))
71	70	3ad2ant1 1164	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ∀𝑝 ∈ 𝑊 ∃𝑢 ∈ 𝑇 𝑝 = (𝐹‘𝑢))
72		dffo3 6600	. 2 ⊢ (𝐹:𝑇–onto→𝑊 ↔ (𝐹:𝑇⟶𝑊 ∧ ∀𝑝 ∈ 𝑊 ∃𝑢 ∈ 𝑇 𝑝 = (𝐹‘𝑢)))
73	6, 71, 72	sylanbrc 579	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–onto→𝑊)

Syntax hints:   → wi 4   ∧ wa 385   ∧ w3a 1108   = wceq 1653  ∃wex 1875   ∈ wcel 2157  ∀wral 3089  ∃wrex 3090  {crab 3093  Vcvv 3385  ⟨cop 4374   class class class wbr 4843   ↦ cmpt 4922  ⟶wf 6097  –onto→wfo 6099  ‘cfv 6101  (class class class)co 6878  1^st c1st 7399  2^nd c2nd 7400  0cc0 10224  ℕ₀cn0 11580  ♯chash 13370  UPGraphcupgr 26315  USPGraphcuspgr 26384  Walkscwlks 26846   WWalksN cwwlksn 27077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ifp 1087  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-2o 7800  df-oadd 7803  df-er 7982  df-map 8097  df-pm 8098  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051  df-cda 9278  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-n0 11581  df-xnn0 11653  df-z 11667  df-uz 11931  df-fz 12581  df-fzo 12721  df-hash 13371  df-word 13535  df-edg 26283  df-uhgr 26293  df-upgr 26317  df-uspgr 26386  df-wlks 26849  df-wwlks 27081  df-wwlksn 27082

This theorem is referenced by:  wlkwwlkbijOLD  27157