Theorem wlknwwlksninj 26638

Description: Lemma 2 for wlknwwlksnbij2 26641. (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
wlknwwlksninj	⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1→𝑊)

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

Allowed substitution hints:   𝑇(𝑝)   𝐹(𝑡,𝑝)   𝑊(𝑝)

Proof of Theorem wlknwwlksninj

Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uspgrupgr 25958	. . 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 26637	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)
6	1, 5	sylan 488	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)
7		fveq2 6150	. . . . . . 7 ⊢ (𝑡 = 𝑣 → (2nd ‘𝑡) = (2nd ‘𝑣))
8		fvex 6160	. . . . . . 7 ⊢ (2nd ‘𝑣) ∈ V
9	7, 4, 8	fvmpt 6240	. . . . . 6 ⊢ (𝑣 ∈ 𝑇 → (𝐹‘𝑣) = (2nd ‘𝑣))
10		fveq2 6150	. . . . . . 7 ⊢ (𝑡 = 𝑤 → (2nd ‘𝑡) = (2nd ‘𝑤))
11		fvex 6160	. . . . . . 7 ⊢ (2nd ‘𝑤) ∈ V
12	10, 4, 11	fvmpt 6240	. . . . . 6 ⊢ (𝑤 ∈ 𝑇 → (𝐹‘𝑤) = (2nd ‘𝑤))
13	9, 12	eqeqan12d 2642	. . . . 5 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (2nd ‘𝑣) = (2nd ‘𝑤)))
14	13	adantl 482	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇)) → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (2nd ‘𝑣) = (2nd ‘𝑤)))
15		fveq2 6150	. . . . . . . . 9 ⊢ (𝑝 = 𝑣 → (1st ‘𝑝) = (1st ‘𝑣))
16	15	fveq2d 6154	. . . . . . . 8 ⊢ (𝑝 = 𝑣 → (#‘(1st ‘𝑝)) = (#‘(1st ‘𝑣)))
17	16	eqeq1d 2628	. . . . . . 7 ⊢ (𝑝 = 𝑣 → ((#‘(1st ‘𝑝)) = 𝑁 ↔ (#‘(1st ‘𝑣)) = 𝑁))
18	17, 2	elrab2 3353	. . . . . 6 ⊢ (𝑣 ∈ 𝑇 ↔ (𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑣)) = 𝑁))
19		fveq2 6150	. . . . . . . . 9 ⊢ (𝑝 = 𝑤 → (1st ‘𝑝) = (1st ‘𝑤))
20	19	fveq2d 6154	. . . . . . . 8 ⊢ (𝑝 = 𝑤 → (#‘(1st ‘𝑝)) = (#‘(1st ‘𝑤)))
21	20	eqeq1d 2628	. . . . . . 7 ⊢ (𝑝 = 𝑤 → ((#‘(1st ‘𝑝)) = 𝑁 ↔ (#‘(1st ‘𝑤)) = 𝑁))
22	21, 2	elrab2 3353	. . . . . 6 ⊢ (𝑤 ∈ 𝑇 ↔ (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑤)) = 𝑁))
23	18, 22	anbi12i 732	. . . . 5 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) ↔ ((𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑣)) = 𝑁) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑤)) = 𝑁)))
24		uspgr2wlkeq2 26406	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑣)) = 𝑁) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑤)) = 𝑁)) → ((2nd ‘𝑣) = (2nd ‘𝑤) → 𝑣 = 𝑤))
25	24	3expb 1263	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑣)) = 𝑁) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑤)) = 𝑁))) → ((2nd ‘𝑣) = (2nd ‘𝑤) → 𝑣 = 𝑤))
26	23, 25	sylan2b 492	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇)) → ((2nd ‘𝑣) = (2nd ‘𝑤) → 𝑣 = 𝑤))
27	14, 26	sylbid 230	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇)) → ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤))
28	27	ralrimivva 2970	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∀𝑣 ∈ 𝑇 ∀𝑤 ∈ 𝑇 ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤))
29		dff13 6467	. 2 ⊢ (𝐹:𝑇–1-1→𝑊 ↔ (𝐹:𝑇⟶𝑊 ∧ ∀𝑣 ∈ 𝑇 ∀𝑤 ∈ 𝑇 ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤)))
30	6, 28, 29	sylanbrc 697	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1→𝑊)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1992  ∀wral 2912  {crab 2916   ↦ cmpt 4678  ⟶wf 5846  –1-1→wf1 5847  ‘cfv 5850  (class class class)co 6605  1^st c1st 7114  2^nd c2nd 7115  ℕ₀cn0 11237  #chash 13054   UPGraph cupgr 25866   USPGraph cuspgr 25931  Walkscwlks 26356   WWalksN cwwlksn 26581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-fzo 12404  df-hash 13055  df-word 13233  df-edg 25835  df-uhgr 25844  df-upgr 25868  df-uspgr 25933  df-wlks 26359  df-wwlks 26585  df-wwlksn 26586

This theorem is referenced by:  wlknwwlksnbij  26640