Theorem wlkiswwlkinj 26246

Description: Lemma 2 for wlkiswwlkbij2 26249. (Contributed by Alexander van der Vekens, 23-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.)

Hypotheses

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

Assertion

Ref	Expression
wlkiswwlkinj	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1→𝑊)

Distinct variable groups:   𝐸,𝑝,𝑡,𝑤   𝑁,𝑝,𝑡,𝑤   𝑃,𝑝,𝑡,𝑤   𝑡,𝑇   𝑉,𝑝,𝑡,𝑤   𝑡,𝑊   𝑤,𝐹   𝑤,𝑇

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

Proof of Theorem wlkiswwlkinj

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkiswwlkbij.t	. . . 4 ⊢ 𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}
2		wlkiswwlkbij.w	. . . 4 ⊢ 𝑊 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}
3		wlkiswwlkbij.f	. . . 4 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡))
4	1, 2, 3	wlkiswwlkfun 26245	. . 3 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)
5	4	3adant1 1072	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)
6		fveq2 6103	. . . . . . 7 ⊢ (𝑡 = 𝑣 → (2nd ‘𝑡) = (2nd ‘𝑣))
7		fvex 6113	. . . . . . 7 ⊢ (2nd ‘𝑣) ∈ V
8	6, 3, 7	fvmpt 6191	. . . . . 6 ⊢ (𝑣 ∈ 𝑇 → (𝐹‘𝑣) = (2nd ‘𝑣))
9		fveq2 6103	. . . . . . 7 ⊢ (𝑡 = 𝑤 → (2nd ‘𝑡) = (2nd ‘𝑤))
10		fvex 6113	. . . . . . 7 ⊢ (2nd ‘𝑤) ∈ V
11	9, 3, 10	fvmpt 6191	. . . . . 6 ⊢ (𝑤 ∈ 𝑇 → (𝐹‘𝑤) = (2nd ‘𝑤))
12	8, 11	eqeqan12d 2626	. . . . 5 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (2nd ‘𝑣) = (2nd ‘𝑤)))
13	12	adantl 481	. . . 4 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇)) → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (2nd ‘𝑣) = (2nd ‘𝑤)))
14		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑝 = 𝑣 → (1st ‘𝑝) = (1st ‘𝑣))
15	14	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑝 = 𝑣 → (#‘(1st ‘𝑝)) = (#‘(1st ‘𝑣)))
16	15	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑝 = 𝑣 → ((#‘(1st ‘𝑝)) = 𝑁 ↔ (#‘(1st ‘𝑣)) = 𝑁))
17		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑝 = 𝑣 → (2nd ‘𝑝) = (2nd ‘𝑣))
18	17	fveq1d 6105	. . . . . . . . 9 ⊢ (𝑝 = 𝑣 → ((2nd ‘𝑝)‘0) = ((2nd ‘𝑣)‘0))
19	18	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑝 = 𝑣 → (((2nd ‘𝑝)‘0) = 𝑃 ↔ ((2nd ‘𝑣)‘0) = 𝑃))
20	16, 19	anbi12d 743	. . . . . . 7 ⊢ (𝑝 = 𝑣 → (((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃) ↔ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)))
21	20, 1	elrab2 3333	. . . . . 6 ⊢ (𝑣 ∈ 𝑇 ↔ (𝑣 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)))
22		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑝 = 𝑤 → (1st ‘𝑝) = (1st ‘𝑤))
23	22	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑝 = 𝑤 → (#‘(1st ‘𝑝)) = (#‘(1st ‘𝑤)))
24	23	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑝 = 𝑤 → ((#‘(1st ‘𝑝)) = 𝑁 ↔ (#‘(1st ‘𝑤)) = 𝑁))
25		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑝 = 𝑤 → (2nd ‘𝑝) = (2nd ‘𝑤))
26	25	fveq1d 6105	. . . . . . . . 9 ⊢ (𝑝 = 𝑤 → ((2nd ‘𝑝)‘0) = ((2nd ‘𝑤)‘0))
27	26	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑝 = 𝑤 → (((2nd ‘𝑝)‘0) = 𝑃 ↔ ((2nd ‘𝑤)‘0) = 𝑃))
28	24, 27	anbi12d 743	. . . . . . 7 ⊢ (𝑝 = 𝑤 → (((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃) ↔ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)))
29	28, 1	elrab2 3333	. . . . . 6 ⊢ (𝑤 ∈ 𝑇 ↔ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)))
30	21, 29	anbi12i 729	. . . . 5 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) ↔ ((𝑣 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃))))
31		3simpb 1052	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0))
32	31	adantr 480	. . . . . 6 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)))) → (𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0))
33		simpl 472	. . . . . . . . 9 ⊢ (((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃) → (#‘(1st ‘𝑣)) = 𝑁)
34	33	anim2i 591	. . . . . . . 8 ⊢ ((𝑣 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) → (𝑣 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘𝑣)) = 𝑁))
35	34	adantr 480	. . . . . . 7 ⊢ (((𝑣 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃))) → (𝑣 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘𝑣)) = 𝑁))
36	35	adantl 481	. . . . . 6 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)))) → (𝑣 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘𝑣)) = 𝑁))
37		simpl 472	. . . . . . . . 9 ⊢ (((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃) → (#‘(1st ‘𝑤)) = 𝑁)
38	37	anim2i 591	. . . . . . . 8 ⊢ ((𝑤 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)) → (𝑤 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘𝑤)) = 𝑁))
39	38	adantl 481	. . . . . . 7 ⊢ (((𝑣 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃))) → (𝑤 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘𝑤)) = 𝑁))
40	39	adantl 481	. . . . . 6 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)))) → (𝑤 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘𝑤)) = 𝑁))
41		usg2wlkeq2 26237	. . . . . 6 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘𝑣)) = 𝑁) ∧ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘𝑤)) = 𝑁)) → ((2nd ‘𝑣) = (2nd ‘𝑤) → 𝑣 = 𝑤))
42	32, 36, 40, 41	syl3anc 1318	. . . . 5 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑣)) = 𝑁 ∧ ((2nd ‘𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)))) → ((2nd ‘𝑣) = (2nd ‘𝑤) → 𝑣 = 𝑤))
43	30, 42	sylan2b 491	. . . 4 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇)) → ((2nd ‘𝑣) = (2nd ‘𝑤) → 𝑣 = 𝑤))
44	13, 43	sylbid 229	. . 3 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇)) → ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤))
45	44	ralrimivva 2954	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ∀𝑣 ∈ 𝑇 ∀𝑤 ∈ 𝑇 ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤))
46		dff13 6416	. 2 ⊢ (𝐹:𝑇–1-1→𝑊 ↔ (𝐹:𝑇⟶𝑊 ∧ ∀𝑣 ∈ 𝑇 ∀𝑤 ∈ 𝑇 ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤)))
47	5, 45, 46	sylanbrc 695	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1→𝑊)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  0cc0 9815  ℕ₀cn0 11169  #chash 12979   USGrph cusg 25859   Walks cwalk 26026   WWalksN cwwlkn 26206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-wwlk 26207  df-wwlkn 26208

This theorem is referenced by:  wlkiswwlkbij  26248