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Theorem wlknwwlksnbij 28930

Description: The mapping (𝑡 ∈ 𝑇 ↦ (2^nd ‘𝑡)) is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length in a simple pseudograph. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 5-Aug-2022.)

**Hypotheses**
Ref	Expression
wlknwwlksnbij.t	⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁}
wlknwwlksnbij.w	⊢ 𝑊 = (𝑁 WWalksN 𝐺)
wlknwwlksnbij.f	⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2^nd ‘𝑡))

**Assertion**
Ref	Expression
wlknwwlksnbij	⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → 𝐹:𝑇–1-1-onto→𝑊)

Distinct variable groups: 𝐺,𝑝,𝑡 𝑁,𝑝,𝑡 𝑡,𝑇 Allowed substitution hints: 𝑇(𝑝) 𝐹(𝑡,𝑝) 𝑊(𝑡,𝑝)

Proof of Theorem wlknwwlksnbij Dummy variable 𝑞 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . 3 ⊢ (𝑝 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑝)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑝))
2	1	wlkswwlksf1o 28921	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝑝 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑝)):(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
3	2	adantr 481	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → (𝑝 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑝)):(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
4		fveqeq2 6871	. . . . 5 ⊢ (𝑞 = (2^nd ‘𝑝) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2^nd ‘𝑝)) = (𝑁 + 1)))
5	4	3ad2ant3 1135	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2^nd ‘𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2^nd ‘𝑝)) = (𝑁 + 1)))
6		wlkcpr 28674	. . . . . . 7 ⊢ (𝑝 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝))
7		wlklenvp1 28663	. . . . . . . 8 ⊢ ((1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝) → (♯‘(2^nd ‘𝑝)) = ((♯‘(1^st ‘𝑝)) + 1))
8		eqeq1 2735	. . . . . . . . . 10 ⊢ ((♯‘(2^nd ‘𝑝)) = ((♯‘(1^st ‘𝑝)) + 1) → ((♯‘(2^nd ‘𝑝)) = (𝑁 + 1) ↔ ((♯‘(1^st ‘𝑝)) + 1) = (𝑁 + 1)))
9		wlkcl 28660	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝) → (♯‘(1^st ‘𝑝)) ∈ ℕ₀)
10	9	nn0cnd 12499	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝) → (♯‘(1^st ‘𝑝)) ∈ ℂ)
11	10	adantr 481	. . . . . . . . . . 11 ⊢ (((1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀)) → (♯‘(1^st ‘𝑝)) ∈ ℂ)
12		nn0cn 12447	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
13	12	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℂ)
14	13	adantl 482	. . . . . . . . . . 11 ⊢ (((1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀)) → 𝑁 ∈ ℂ)
15		1cnd 11174	. . . . . . . . . . 11 ⊢ (((1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀)) → 1 ∈ ℂ)
16	11, 14, 15	addcan2d 11383	. . . . . . . . . 10 ⊢ (((1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀)) → (((♯‘(1^st ‘𝑝)) + 1) = (𝑁 + 1) ↔ (♯‘(1^st ‘𝑝)) = 𝑁))
17	8, 16	sylan9bbr 511	. . . . . . . . 9 ⊢ ((((1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘(2^nd ‘𝑝)) = ((♯‘(1^st ‘𝑝)) + 1)) → ((♯‘(2^nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1^st ‘𝑝)) = 𝑁))
18	17	exp31 420	. . . . . . . 8 ⊢ ((1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → ((♯‘(2^nd ‘𝑝)) = ((♯‘(1^st ‘𝑝)) + 1) → ((♯‘(2^nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1^st ‘𝑝)) = 𝑁))))
19	7, 18	mpid 44	. . . . . . 7 ⊢ ((1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → ((♯‘(2^nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1^st ‘𝑝)) = 𝑁)))
20	6, 19	sylbi 216	. . . . . 6 ⊢ (𝑝 ∈ (Walks‘𝐺) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → ((♯‘(2^nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1^st ‘𝑝)) = 𝑁)))
21	20	impcom 408	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ 𝑝 ∈ (Walks‘𝐺)) → ((♯‘(2^nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1^st ‘𝑝)) = 𝑁))
22	21	3adant3 1132	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2^nd ‘𝑝)) → ((♯‘(2^nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1^st ‘𝑝)) = 𝑁))
23	5, 22	bitrd 278	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2^nd ‘𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(1^st ‘𝑝)) = 𝑁))
24	1, 3, 23	f1oresrab 7093	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → ((𝑝 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁}):{𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁}–1-1-onto→{𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
25		wlknwwlksnbij.f	. . . 4 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2^nd ‘𝑡))
26		wlknwwlksnbij.t	. . . . . . 7 ⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁}
27	26	mpteq1i 5221	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ (2^nd ‘𝑡)) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ↦ (2^nd ‘𝑡))
28		ssrab2 4057	. . . . . . 7 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ⊆ (Walks‘𝐺)
29		resmpt 6011	. . . . . . 7 ⊢ ({𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ⊆ (Walks‘𝐺) → ((𝑡 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁}) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ↦ (2^nd ‘𝑡)))
30	28, 29	ax-mp 5	. . . . . 6 ⊢ ((𝑡 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁}) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ↦ (2^nd ‘𝑡))
31		fveq2 6862	. . . . . . . 8 ⊢ (𝑡 = 𝑝 → (2^nd ‘𝑡) = (2^nd ‘𝑝))
32	31	cbvmptv 5238	. . . . . . 7 ⊢ (𝑡 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑡)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑝))
33	32	reseq1i 5953	. . . . . 6 ⊢ ((𝑡 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁}) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁})
34	27, 30, 33	3eqtr2i 2765	. . . . 5 ⊢ (𝑡 ∈ 𝑇 ↦ (2^nd ‘𝑡)) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁})
35	34	a1i 11	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → (𝑡 ∈ 𝑇 ↦ (2^nd ‘𝑡)) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁}))
36	25, 35	eqtrid 2783	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → 𝐹 = ((𝑝 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁}))
37	26	a1i 11	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁})
38		wlknwwlksnbij.w	. . . 4 ⊢ 𝑊 = (𝑁 WWalksN 𝐺)
39		wwlksn 28879	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 WWalksN 𝐺) = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
40	39	adantl 482	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → (𝑁 WWalksN 𝐺) = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
41	38, 40	eqtrid 2783	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → 𝑊 = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
42	36, 37, 41	f1oeq123d 6798	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → (𝐹:𝑇–1-1-onto→𝑊 ↔ ((𝑝 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁}):{𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁}–1-1-onto→{𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)}))
43	24, 42	mpbird 256	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → 𝐹:𝑇–1-1-onto→𝑊)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {crab 3418 ⊆ wss 3928 class class class wbr 5125 ↦ cmpt 5208 ↾ cres 5655 –1-1-onto→wf1o 6515 ‘cfv 6516 (class class class)co 7377 1^st c1st 7939 2^nd c2nd 7940 ℂcc 11073 1c1 11076 + caddc 11078 ℕ₀cn0 12437 ♯chash 14255 USPGraphcuspgr 28196 Walkscwlks 28641 WWalkscwwlks 28867 WWalksN cwwlksn 28868

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-int 4928 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-om 7823 df-1st 7941 df-2nd 7942 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-oadd 8436 df-er 8670 df-map 8789 df-pm 8790 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-dju 9861 df-card 9899 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-nn 12178 df-2 12240 df-n0 12438 df-xnn0 12510 df-z 12524 df-uz 12788 df-fz 13450 df-fzo 13593 df-hash 14256 df-word 14430 df-edg 28096 df-uhgr 28106 df-upgr 28130 df-uspgr 28198 df-wlks 28644 df-wwlks 28872 df-wwlksn 28873

This theorem is referenced by: wlknwwlksnen 28931 wlksnwwlknvbij 28950

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