Theorem wlknwwlksninj 26843

Description: Lemma 2 for wlknwwlksnbij2 26846. (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 26116	. . 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 26842	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)
6	1, 5	sylan 487	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)
7		fveq2 6229	. . . . . . 7 ⊢ (𝑡 = 𝑣 → (2nd ‘𝑡) = (2nd ‘𝑣))
8		fvex 6239	. . . . . . 7 ⊢ (2nd ‘𝑣) ∈ V
9	7, 4, 8	fvmpt 6321	. . . . . 6 ⊢ (𝑣 ∈ 𝑇 → (𝐹‘𝑣) = (2nd ‘𝑣))
10		fveq2 6229	. . . . . . 7 ⊢ (𝑡 = 𝑤 → (2nd ‘𝑡) = (2nd ‘𝑤))
11		fvex 6239	. . . . . . 7 ⊢ (2nd ‘𝑤) ∈ V
12	10, 4, 11	fvmpt 6321	. . . . . 6 ⊢ (𝑤 ∈ 𝑇 → (𝐹‘𝑤) = (2nd ‘𝑤))
13	9, 12	eqeqan12d 2667	. . . . 5 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (2nd ‘𝑣) = (2nd ‘𝑤)))
14	13	adantl 481	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇)) → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (2nd ‘𝑣) = (2nd ‘𝑤)))
15		fveq2 6229	. . . . . . . . 9 ⊢ (𝑝 = 𝑣 → (1st ‘𝑝) = (1st ‘𝑣))
16	15	fveq2d 6233	. . . . . . . 8 ⊢ (𝑝 = 𝑣 → (#‘(1st ‘𝑝)) = (#‘(1st ‘𝑣)))
17	16	eqeq1d 2653	. . . . . . 7 ⊢ (𝑝 = 𝑣 → ((#‘(1st ‘𝑝)) = 𝑁 ↔ (#‘(1st ‘𝑣)) = 𝑁))
18	17, 2	elrab2 3399	. . . . . 6 ⊢ (𝑣 ∈ 𝑇 ↔ (𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑣)) = 𝑁))
19		fveq2 6229	. . . . . . . . 9 ⊢ (𝑝 = 𝑤 → (1st ‘𝑝) = (1st ‘𝑤))
20	19	fveq2d 6233	. . . . . . . 8 ⊢ (𝑝 = 𝑤 → (#‘(1st ‘𝑝)) = (#‘(1st ‘𝑤)))
21	20	eqeq1d 2653	. . . . . . 7 ⊢ (𝑝 = 𝑤 → ((#‘(1st ‘𝑝)) = 𝑁 ↔ (#‘(1st ‘𝑤)) = 𝑁))
22	21, 2	elrab2 3399	. . . . . 6 ⊢ (𝑤 ∈ 𝑇 ↔ (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑤)) = 𝑁))
23	18, 22	anbi12i 733	. . . . 5 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) ↔ ((𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑣)) = 𝑁) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑤)) = 𝑁)))
24		uspgr2wlkeq2 26599	. . . . . 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 1285	. . . . 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 491	. . . 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 3000	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∀𝑣 ∈ 𝑇 ∀𝑤 ∈ 𝑇 ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤))
29		dff13 6552	. 2 ⊢ (𝐹:𝑇–1-1→𝑊 ↔ (𝐹:𝑇⟶𝑊 ∧ ∀𝑣 ∈ 𝑇 ∀𝑤 ∈ 𝑇 ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤)))
30	6, 28, 29	sylanbrc 699	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1→𝑊)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  {crab 2945   ↦ cmpt 4762  ⟶wf 5922  –1-1→wf1 5923  ‘cfv 5926  (class class class)co 6690  1^st c1st 7208  2^nd c2nd 7209  ℕ₀cn0 11330  #chash 13157  UPGraphcupgr 26020  USPGraphcuspgr 26088  Walkscwlks 26548   WWalksN cwwlksn 26774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-edg 25985  df-uhgr 25998  df-upgr 26022  df-uspgr 26090  df-wlks 26551  df-wwlks 26778  df-wwlksn 26779

This theorem is referenced by:  wlknwwlksnbij  26845