Theorem wlknwwlksnsur 26776

Description: Lemma 3 for wlknwwlksnbij2 26778. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)

Hypotheses

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

Assertion

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

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

Allowed substitution hint:   𝐹(𝑡)

Proof of Theorem wlknwwlksnsur

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

Step	Hyp	Ref	Expression
1		uspgrupgr 26071	. . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )
2		wlknwwlksnbij.t	. . . 4 ⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st ‘𝑝)) = 𝑁}
3		wlknwwlksnbij.w	. . . 4 ⊢ 𝑊 = (𝑁 WWalksN 𝐺)
4		wlknwwlksnbij.f	. . . 4 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡))
5	2, 3, 4	wlknwwlksnfun 26774	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)
6	1, 5	sylan 488	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)
7	3	eleq2i 2693	. . . 4 ⊢ (𝑝 ∈ 𝑊 ↔ 𝑝 ∈ (𝑁 WWalksN 𝐺))
8		wlklnwwlkn 26770	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 𝑁) ↔ 𝑝 ∈ (𝑁 WWalksN 𝐺)))
9	8	adantr 481	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (∃𝑓(𝑓(Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 𝑁) ↔ 𝑝 ∈ (𝑁 WWalksN 𝐺)))
10		df-br 4654	. . . . . . . . . . . 12 ⊢ (𝑓(Walks‘𝐺)𝑝 ↔ ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺))
11		vex 3203	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
12		vex 3203	. . . . . . . . . . . . . . . 16 ⊢ 𝑝 ∈ V
13	11, 12	op1st 7176	. . . . . . . . . . . . . . 15 ⊢ (1st ‘⟨𝑓, 𝑝⟩) = 𝑓
14	13	eqcomi 2631	. . . . . . . . . . . . . 14 ⊢ 𝑓 = (1st ‘⟨𝑓, 𝑝⟩)
15	14	fveq2i 6194	. . . . . . . . . . . . 13 ⊢ (#‘𝑓) = (#‘(1st ‘⟨𝑓, 𝑝⟩))
16	15	eqeq1i 2627	. . . . . . . . . . . 12 ⊢ ((#‘𝑓) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁)
17		elex 3212	. . . . . . . . . . . . . 14 ⊢ (⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) → ⟨𝑓, 𝑝⟩ ∈ V)
18		eleq1 2689	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → (𝑢 ∈ (Walks‘𝐺) ↔ ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺)))
19	18	biimparc 504	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → 𝑢 ∈ (Walks‘𝐺))
20	19	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → 𝑢 ∈ (Walks‘𝐺))
21		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → (1st ‘𝑢) = (1st ‘⟨𝑓, 𝑝⟩))
22	21	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → (#‘(1st ‘𝑢)) = (#‘(1st ‘⟨𝑓, 𝑝⟩)))
23	22	eqeq1d 2624	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → ((#‘(1st ‘𝑢)) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁))
24	23	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → ((#‘(1st ‘𝑢)) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁))
25	24	biimpar 502	. . . . . . . . . . . . . . . 16 ⊢ (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → (#‘(1st ‘𝑢)) = 𝑁)
26		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → (2nd ‘𝑢) = (2nd ‘⟨𝑓, 𝑝⟩))
27	11, 12	op2nd 7177	. . . . . . . . . . . . . . . . . . 19 ⊢ (2nd ‘⟨𝑓, 𝑝⟩) = 𝑝
28	26, 27	syl6req 2673	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = ⟨𝑓, 𝑝⟩ → 𝑝 = (2nd ‘𝑢))
29	28	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → 𝑝 = (2nd ‘𝑢))
30	29	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → 𝑝 = (2nd ‘𝑢))
31	20, 25, 30	jca31 557	. . . . . . . . . . . . . . 15 ⊢ (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑢)) = 𝑁) ∧ 𝑝 = (2nd ‘𝑢)))
32	31	ex 450	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → ((#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 → ((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑢)) = 𝑁) ∧ 𝑝 = (2nd ‘𝑢))))
33	17, 32	spcimedv 3292	. . . . . . . . . . . . 13 ⊢ (⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) → ((#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑢)) = 𝑁) ∧ 𝑝 = (2nd ‘𝑢))))
34	33	imp 445	. . . . . . . . . . . 12 ⊢ ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑢)) = 𝑁) ∧ 𝑝 = (2nd ‘𝑢)))
35	10, 16, 34	syl2anb 496	. . . . . . . . . . 11 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 𝑁) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑢)) = 𝑁) ∧ 𝑝 = (2nd ‘𝑢)))
36	35	exlimiv 1858	. . . . . . . . . 10 ⊢ (∃𝑓(𝑓(Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 𝑁) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑢)) = 𝑁) ∧ 𝑝 = (2nd ‘𝑢)))
37	9, 36	syl6bir 244	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑝 ∈ (𝑁 WWalksN 𝐺) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑢)) = 𝑁) ∧ 𝑝 = (2nd ‘𝑢))))
38	37	imp 445	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑢)) = 𝑁) ∧ 𝑝 = (2nd ‘𝑢)))
39		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑢 → (1st ‘𝑝) = (1st ‘𝑢))
40	39	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑢 → (#‘(1st ‘𝑝)) = (#‘(1st ‘𝑢)))
41	40	eqeq1d 2624	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑢 → ((#‘(1st ‘𝑝)) = 𝑁 ↔ (#‘(1st ‘𝑢)) = 𝑁))
42	41	elrab 3363	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st ‘𝑝)) = 𝑁} ↔ (𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑢)) = 𝑁))
43	42	anbi1i 731	. . . . . . . . 9 ⊢ ((𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st ‘𝑝)) = 𝑁} ∧ 𝑝 = (2nd ‘𝑢)) ↔ ((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑢)) = 𝑁) ∧ 𝑝 = (2nd ‘𝑢)))
44	43	exbii 1774	. . . . . . . 8 ⊢ (∃𝑢(𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st ‘𝑝)) = 𝑁} ∧ 𝑝 = (2nd ‘𝑢)) ↔ ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑢)) = 𝑁) ∧ 𝑝 = (2nd ‘𝑢)))
45	38, 44	sylibr 224	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢(𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st ‘𝑝)) = 𝑁} ∧ 𝑝 = (2nd ‘𝑢)))
46		df-rex 2918	. . . . . . 7 ⊢ (∃𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st ‘𝑝)) = 𝑁}𝑝 = (2nd ‘𝑢) ↔ ∃𝑢(𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st ‘𝑝)) = 𝑁} ∧ 𝑝 = (2nd ‘𝑢)))
47	45, 46	sylibr 224	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st ‘𝑝)) = 𝑁}𝑝 = (2nd ‘𝑢))
48	2	rexeqi 3143	. . . . . 6 ⊢ (∃𝑢 ∈ 𝑇 𝑝 = (2nd ‘𝑢) ↔ ∃𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st ‘𝑝)) = 𝑁}𝑝 = (2nd ‘𝑢))
49	47, 48	sylibr 224	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢 ∈ 𝑇 𝑝 = (2nd ‘𝑢))
50		fveq2 6191	. . . . . . . 8 ⊢ (𝑡 = 𝑢 → (2nd ‘𝑡) = (2nd ‘𝑢))
51		fvex 6201	. . . . . . . 8 ⊢ (2nd ‘𝑢) ∈ V
52	50, 4, 51	fvmpt 6282	. . . . . . 7 ⊢ (𝑢 ∈ 𝑇 → (𝐹‘𝑢) = (2nd ‘𝑢))
53	52	eqeq2d 2632	. . . . . 6 ⊢ (𝑢 ∈ 𝑇 → (𝑝 = (𝐹‘𝑢) ↔ 𝑝 = (2nd ‘𝑢)))
54	53	rexbiia 3040	. . . . 5 ⊢ (∃𝑢 ∈ 𝑇 𝑝 = (𝐹‘𝑢) ↔ ∃𝑢 ∈ 𝑇 𝑝 = (2nd ‘𝑢))
55	49, 54	sylibr 224	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢 ∈ 𝑇 𝑝 = (𝐹‘𝑢))
56	7, 55	sylan2b 492	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ 𝑊) → ∃𝑢 ∈ 𝑇 𝑝 = (𝐹‘𝑢))
57	56	ralrimiva 2966	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∀𝑝 ∈ 𝑊 ∃𝑢 ∈ 𝑇 𝑝 = (𝐹‘𝑢))
58		dffo3 6374	. 2 ⊢ (𝐹:𝑇–onto→𝑊 ↔ (𝐹:𝑇⟶𝑊 ∧ ∀𝑝 ∈ 𝑊 ∃𝑢 ∈ 𝑇 𝑝 = (𝐹‘𝑢)))
59	6, 57, 58	sylanbrc 698	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–onto→𝑊)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200  ⟨cop 4183   class class class wbr 4653   ↦ cmpt 4729  ⟶wf 5884  –onto→wfo 5886  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  ℕ₀cn0 11292  #chash 13117   UPGraph cupgr 25975   USPGraph cuspgr 26043  Walkscwlks 26492   WWalksN cwwlksn 26718

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-ifp 1013  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-rmo 2920  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-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-cda 8990  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-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-uspgr 26045  df-wlks 26495  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  wlknwwlksnbij  26777