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Theorem wlknwwlksnbij 29142

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 2733	. . 3 ⊢ (𝑝 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑝)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑝))
2	1	wlkswwlksf1o 29133	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝑝 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑝)):(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
3	2	adantr 482	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → (𝑝 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑝)):(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
4		fveqeq2 6901	. . . . 5 ⊢ (𝑞 = (2^nd ‘𝑝) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2^nd ‘𝑝)) = (𝑁 + 1)))
5	4	3ad2ant3 1136	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2^nd ‘𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2^nd ‘𝑝)) = (𝑁 + 1)))
6		wlkcpr 28886	. . . . . . 7 ⊢ (𝑝 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝))
7		wlklenvp1 28875	. . . . . . . 8 ⊢ ((1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝) → (♯‘(2^nd ‘𝑝)) = ((♯‘(1^st ‘𝑝)) + 1))
8		eqeq1 2737	. . . . . . . . . 10 ⊢ ((♯‘(2^nd ‘𝑝)) = ((♯‘(1^st ‘𝑝)) + 1) → ((♯‘(2^nd ‘𝑝)) = (𝑁 + 1) ↔ ((♯‘(1^st ‘𝑝)) + 1) = (𝑁 + 1)))
9		wlkcl 28872	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝) → (♯‘(1^st ‘𝑝)) ∈ ℕ₀)
10	9	nn0cnd 12534	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝) → (♯‘(1^st ‘𝑝)) ∈ ℂ)
11	10	adantr 482	. . . . . . . . . . 11 ⊢ (((1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀)) → (♯‘(1^st ‘𝑝)) ∈ ℂ)
12		nn0cn 12482	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
13	12	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℂ)
14	13	adantl 483	. . . . . . . . . . 11 ⊢ (((1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀)) → 𝑁 ∈ ℂ)
15		1cnd 11209	. . . . . . . . . . 11 ⊢ (((1^st ‘𝑝)(Walks‘𝐺)(2^nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀)) → 1 ∈ ℂ)
16	11, 14, 15	addcan2d 11418	. . . . . . . . . 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 512	. . . . . . . . 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 421	. . . . . . . 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 409	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ 𝑝 ∈ (Walks‘𝐺)) → ((♯‘(2^nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1^st ‘𝑝)) = 𝑁))
22	21	3adant3 1133	. . . 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 279	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2^nd ‘𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(1^st ‘𝑝)) = 𝑁))
24	1, 3, 23	f1oresrab 7125	. 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 5245	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ (2^nd ‘𝑡)) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ↦ (2^nd ‘𝑡))
28		ssrab2 4078	. . . . . . 7 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ⊆ (Walks‘𝐺)
29		resmpt 6038	. . . . . . 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 6892	. . . . . . . 8 ⊢ (𝑡 = 𝑝 → (2^nd ‘𝑡) = (2^nd ‘𝑝))
32	31	cbvmptv 5262	. . . . . . 7 ⊢ (𝑡 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑡)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2^nd ‘𝑝))
33	32	reseq1i 5978	. . . . . 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 2767	. . . . 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 2785	. . 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 29091	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 WWalksN 𝐺) = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
40	39	adantl 483	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → (𝑁 WWalksN 𝐺) = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
41	38, 40	eqtrid 2785	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → 𝑊 = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
42	36, 37, 41	f1oeq123d 6828	. 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 257	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → 𝐹:𝑇–1-1-onto→𝑊)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {crab 3433 ⊆ wss 3949 class class class wbr 5149 ↦ cmpt 5232 ↾ cres 5679 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 2^nd c2nd 7974 ℂcc 11108 1c1 11111 + caddc 11113 ℕ₀cn0 12472 ♯chash 14290 USPGraphcuspgr 28408 Walkscwlks 28853 WWalkscwwlks 29079 WWalksN cwwlksn 29080

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-edg 28308 df-uhgr 28318 df-upgr 28342 df-uspgr 28410 df-wlks 28856 df-wwlks 29084 df-wwlksn 29085

This theorem is referenced by: wlknwwlksnen 29143 wlksnwwlknvbij 29162

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