Theorem wlkwwlkinj 26782

Description: Lemma 2 for wlkwwlkbij2 26785. (Contributed by Alexander van der Vekens, 23-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 16-Apr-2021.)

Hypotheses

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

Assertion

Ref	Expression
wlkwwlkinj	⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1→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

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

Proof of Theorem wlkwwlkinj

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uspgrupgr 26071	. . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )
2		wlkwwlkbij.t	. . . 4 ⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}
3		wlkwwlkbij.w	. . . 4 ⊢ 𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
4		wlkwwlkbij.f	. . . 4 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡))
5	2, 3, 4	wlkwwlkfun 26781	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)
6	1, 5	syl3an1 1359	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)
7		fveq2 6191	. . . . . . 7 ⊢ (𝑡 = 𝑣 → (2nd ‘𝑡) = (2nd ‘𝑣))
8		fvex 6201	. . . . . . 7 ⊢ (2nd ‘𝑣) ∈ V
9	7, 4, 8	fvmpt 6282	. . . . . 6 ⊢ (𝑣 ∈ 𝑇 → (𝐹‘𝑣) = (2nd ‘𝑣))
10		fveq2 6191	. . . . . . 7 ⊢ (𝑡 = 𝑤 → (2nd ‘𝑡) = (2nd ‘𝑤))
11		fvex 6201	. . . . . . 7 ⊢ (2nd ‘𝑤) ∈ V
12	10, 4, 11	fvmpt 6282	. . . . . 6 ⊢ (𝑤 ∈ 𝑇 → (𝐹‘𝑤) = (2nd ‘𝑤))
13	9, 12	eqeqan12d 2638	. . . . 5 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (2nd ‘𝑣) = (2nd ‘𝑤)))
14	13	adantl 482	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇)) → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (2nd ‘𝑣) = (2nd ‘𝑤)))
15		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑝 = 𝑣 → (1st ‘𝑝) = (1st ‘𝑣))
16	15	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑝 = 𝑣 → (#‘(1st ‘𝑝)) = (#‘(1st ‘𝑣)))
17	16	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑝 = 𝑣 → ((#‘(1st ‘𝑝)) = 𝑁 ↔ (#‘(1st ‘𝑣)) = 𝑁))
18		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑝 = 𝑣 → (2nd ‘𝑝) = (2nd ‘𝑣))
19	18	fveq1d 6193	. . . . . . . . 9 ⊢ (𝑝 = 𝑣 → ((2nd ‘𝑝)‘0) = ((2nd ‘𝑣)‘0))
20	19	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑝 = 𝑣 → (((2nd ‘𝑝)‘0) = 𝑃 ↔ ((2nd ‘𝑣)‘0) = 𝑃))
21	17, 20	anbi12d 747	. . . . . . 7 ⊢ (𝑝 = 𝑣 → (((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃) ↔ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)))
22	21, 2	elrab2 3366	. . . . . 6 ⊢ (𝑣 ∈ 𝑇 ↔ (𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)))
23		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑝 = 𝑤 → (1st ‘𝑝) = (1st ‘𝑤))
24	23	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑝 = 𝑤 → (#‘(1st ‘𝑝)) = (#‘(1st ‘𝑤)))
25	24	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑝 = 𝑤 → ((#‘(1st ‘𝑝)) = 𝑁 ↔ (#‘(1st ‘𝑤)) = 𝑁))
26		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑝 = 𝑤 → (2nd ‘𝑝) = (2nd ‘𝑤))
27	26	fveq1d 6193	. . . . . . . . 9 ⊢ (𝑝 = 𝑤 → ((2nd ‘𝑝)‘0) = ((2nd ‘𝑤)‘0))
28	27	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑝 = 𝑤 → (((2nd ‘𝑝)‘0) = 𝑃 ↔ ((2nd ‘𝑤)‘0) = 𝑃))
29	25, 28	anbi12d 747	. . . . . . 7 ⊢ (𝑝 = 𝑤 → (((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃) ↔ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)))
30	29, 2	elrab2 3366	. . . . . 6 ⊢ (𝑤 ∈ 𝑇 ↔ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)))
31	22, 30	anbi12i 733	. . . . 5 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) ↔ ((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃))))
32		3simpb 1059	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0))
33	32	adantr 481	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)))) → (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0))
34		simpl 473	. . . . . . . . 9 ⊢ (((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃) → (#‘(1st ‘𝑣)) = 𝑁)
35	34	anim2i 593	. . . . . . . 8 ⊢ ((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) → (𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑣)) = 𝑁))
36	35	adantr 481	. . . . . . 7 ⊢ (((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃))) → (𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑣)) = 𝑁))
37	36	adantl 482	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)))) → (𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑣)) = 𝑁))
38		simpl 473	. . . . . . . . 9 ⊢ (((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃) → (#‘(1st ‘𝑤)) = 𝑁)
39	38	anim2i 593	. . . . . . . 8 ⊢ ((𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)) → (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑤)) = 𝑁))
40	39	adantl 482	. . . . . . 7 ⊢ (((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃))) → (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑤)) = 𝑁))
41	40	adantl 482	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)))) → (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑤)) = 𝑁))
42		uspgr2wlkeq2 26543	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑣)) = 𝑁) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑤)) = 𝑁)) → ((2nd ‘𝑣) = (2nd ‘𝑤) → 𝑣 = 𝑤))
43	33, 37, 41, 42	syl3anc 1326	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)))) → ((2nd ‘𝑣) = (2nd ‘𝑤) → 𝑣 = 𝑤))
44	31, 43	sylan2b 492	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇)) → ((2nd ‘𝑣) = (2nd ‘𝑤) → 𝑣 = 𝑤))
45	14, 44	sylbid 230	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇)) → ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤))
46	45	ralrimivva 2971	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ∀𝑣 ∈ 𝑇 ∀𝑤 ∈ 𝑇 ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤))
47		dff13 6512	. 2 ⊢ (𝐹:𝑇–1-1→𝑊 ↔ (𝐹:𝑇⟶𝑊 ∧ ∀𝑣 ∈ 𝑇 ∀𝑤 ∈ 𝑇 ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤)))
48	6, 46, 47	sylanbrc 698	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1→𝑊)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916   ↦ cmpt 4729  ⟶wf 5884  –1-1→wf1 5885  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  0cc0 9936  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:  wlkwwlkbij  26784