Theorem wlkwwlkinj 27005

Description: Lemma 2 for wlkwwlkbij2 27008. (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 26270	. . 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 27004	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)
6	1, 5	syl3an1 1167	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)
7		fveq2 6352	. . . . . . 7 ⊢ (𝑡 = 𝑣 → (2nd ‘𝑡) = (2nd ‘𝑣))
8		fvex 6362	. . . . . . 7 ⊢ (2nd ‘𝑣) ∈ V
9	7, 4, 8	fvmpt 6444	. . . . . 6 ⊢ (𝑣 ∈ 𝑇 → (𝐹‘𝑣) = (2nd ‘𝑣))
10		fveq2 6352	. . . . . . 7 ⊢ (𝑡 = 𝑤 → (2nd ‘𝑡) = (2nd ‘𝑤))
11		fvex 6362	. . . . . . 7 ⊢ (2nd ‘𝑤) ∈ V
12	10, 4, 11	fvmpt 6444	. . . . . 6 ⊢ (𝑤 ∈ 𝑇 → (𝐹‘𝑤) = (2nd ‘𝑤))
13	9, 12	eqeqan12d 2776	. . . . 5 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (2nd ‘𝑣) = (2nd ‘𝑤)))
14	13	adantl 473	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇)) → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (2nd ‘𝑣) = (2nd ‘𝑤)))
15		fveq2 6352	. . . . . . . . . 10 ⊢ (𝑝 = 𝑣 → (1st ‘𝑝) = (1st ‘𝑣))
16	15	fveq2d 6356	. . . . . . . . 9 ⊢ (𝑝 = 𝑣 → (♯‘(1st ‘𝑝)) = (♯‘(1st ‘𝑣)))
17	16	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑝 = 𝑣 → ((♯‘(1st ‘𝑝)) = 𝑁 ↔ (♯‘(1st ‘𝑣)) = 𝑁))
18		fveq2 6352	. . . . . . . . . 10 ⊢ (𝑝 = 𝑣 → (2nd ‘𝑝) = (2nd ‘𝑣))
19	18	fveq1d 6354	. . . . . . . . 9 ⊢ (𝑝 = 𝑣 → ((2nd ‘𝑝)‘0) = ((2nd ‘𝑣)‘0))
20	19	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑝 = 𝑣 → (((2nd ‘𝑝)‘0) = 𝑃 ↔ ((2nd ‘𝑣)‘0) = 𝑃))
21	17, 20	anbi12d 749	. . . . . . 7 ⊢ (𝑝 = 𝑣 → (((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃) ↔ ((♯‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)))
22	21, 2	elrab2 3507	. . . . . 6 ⊢ (𝑣 ∈ 𝑇 ↔ (𝑣 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)))
23		fveq2 6352	. . . . . . . . . 10 ⊢ (𝑝 = 𝑤 → (1st ‘𝑝) = (1st ‘𝑤))
24	23	fveq2d 6356	. . . . . . . . 9 ⊢ (𝑝 = 𝑤 → (♯‘(1st ‘𝑝)) = (♯‘(1st ‘𝑤)))
25	24	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑝 = 𝑤 → ((♯‘(1st ‘𝑝)) = 𝑁 ↔ (♯‘(1st ‘𝑤)) = 𝑁))
26		fveq2 6352	. . . . . . . . . 10 ⊢ (𝑝 = 𝑤 → (2nd ‘𝑝) = (2nd ‘𝑤))
27	26	fveq1d 6354	. . . . . . . . 9 ⊢ (𝑝 = 𝑤 → ((2nd ‘𝑝)‘0) = ((2nd ‘𝑤)‘0))
28	27	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑝 = 𝑤 → (((2nd ‘𝑝)‘0) = 𝑃 ↔ ((2nd ‘𝑤)‘0) = 𝑃))
29	25, 28	anbi12d 749	. . . . . . 7 ⊢ (𝑝 = 𝑤 → (((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃) ↔ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)))
30	29, 2	elrab2 3507	. . . . . 6 ⊢ (𝑤 ∈ 𝑇 ↔ (𝑤 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)))
31	22, 30	anbi12i 735	. . . . 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 1145	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0))
33	32	adantr 472	. . . . . 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 474	. . . . . . . . 9 ⊢ (((♯‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃) → (♯‘(1st ‘𝑣)) = 𝑁)
35	34	anim2i 594	. . . . . . . 8 ⊢ ((𝑣 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) → (𝑣 ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘𝑣)) = 𝑁))
36	35	adantr 472	. . . . . . 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 473	. . . . . 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 474	. . . . . . . . 9 ⊢ (((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃) → (♯‘(1st ‘𝑤)) = 𝑁)
39	38	anim2i 594	. . . . . . . 8 ⊢ ((𝑤 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)) → (𝑤 ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘𝑤)) = 𝑁))
40	39	adantl 473	. . . . . . 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 473	. . . . . 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 26753	. . . . . 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 1477	. . . . 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 493	. . . 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 3109	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ∀𝑣 ∈ 𝑇 ∀𝑤 ∈ 𝑇 ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤))
47		dff13 6675	. 2 ⊢ (𝐹:𝑇–1-1→𝑊 ↔ (𝐹:𝑇⟶𝑊 ∧ ∀𝑣 ∈ 𝑇 ∀𝑤 ∈ 𝑇 ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤)))
48	6, 46, 47	sylanbrc 701	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1→𝑊)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∀wral 3050  {crab 3054   ↦ cmpt 4881  ⟶wf 6045  –1-1→wf1 6046  ‘cfv 6049  (class class class)co 6813  1^st c1st 7331  2^nd c2nd 7332  0cc0 10128  ℕ₀cn0 11484  ♯chash 13311  UPGraphcupgr 26174  USPGraphcuspgr 26242  Walkscwlks 26702   WWalksN cwwlksn 26929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1051  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-n0 11485  df-xnn0 11556  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660  df-hash 13312  df-word 13485  df-edg 26139  df-uhgr 26152  df-upgr 26176  df-uspgr 26244  df-wlks 26705  df-wwlks 26933  df-wwlksn 26934

This theorem is referenced by:  wlkwwlkbij  27007