Theorem wlkswwlksf1o 30495

Description: The mapping of (ordinary) walks to their sequences of vertices is a bijection in a simple pseudograph. (Contributed by AV, 6-May-2021.)

Hypothesis

Ref	Expression
wlkswwlksf1o.f	⊢ 𝐹 = (𝑤 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑤))

Assertion

Ref	Expression
wlkswwlksf1o	⊢ (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)⟶1-1onto(WWalks‘𝐺))

Distinct variable group:   𝑤,𝐺

Allowed substitution hint:   𝐹(𝑤)

Proof of Theorem wlkswwlksf1o

Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6918	. . . . . 6 ⊢ (1st ‘𝑤) ∈ V
2		breq1 5109	. . . . . 6 ⊢ (𝑓 = (1st ‘𝑤) → (𝑓(Walks‘𝐺)(2nd ‘𝑤) ↔ (1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤)))
3	1, 2	spcev 3564	. . . . 5 ⊢ ((1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤) → ∃𝑓 𝑓(Walks‘𝐺)(2nd ‘𝑤))
4		wlkiswwlks 30492	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (∃𝑓 𝑓(Walks‘𝐺)(2nd ‘𝑤) ↔ (2nd ‘𝑤) ∈ (WWalks‘𝐺)))
5	3, 4	imbitrid 247	. . . 4 ⊢ (𝐺 ∈ USPGraph → ((1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤) → (2nd ‘𝑤) ∈ (WWalks‘𝐺)))
6		wlkcpr 30236	. . . . 5 ⊢ (𝑤 ∈ (Walks‘𝐺) ↔ (1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤))
7	6	biimpi 219	. . . 4 ⊢ (𝑤 ∈ (Walks‘𝐺) → (1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤))
8	5, 7	impel 515	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (Walks‘𝐺)) → (2nd ‘𝑤) ∈ (WWalks‘𝐺))
9		wlkswwlksf1o.f	. . 3 ⊢ 𝐹 = (𝑤 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑤))
10	8, 9	fmptd 7135	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺))
11		simpr 490	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺))
12		fveq2 6905	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (2nd ‘𝑤) = (2nd ‘𝑥))
13		id 23	. . . . . . . . 9 ⊢ (𝑥 ∈ (Walks‘𝐺) → 𝑥 ∈ (Walks‘𝐺))
14		fvexd 6920	. . . . . . . . 9 ⊢ (𝑥 ∈ (Walks‘𝐺) → (2nd ‘𝑥) ∈ V)
15	9, 12, 13, 14	fvmptd3 7037	. . . . . . . 8 ⊢ (𝑥 ∈ (Walks‘𝐺) → (𝐹‘𝑥) = (2nd ‘𝑥))
16		fveq2 6905	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (2nd ‘𝑤) = (2nd ‘𝑦))
17		id 23	. . . . . . . . 9 ⊢ (𝑦 ∈ (Walks‘𝐺) → 𝑦 ∈ (Walks‘𝐺))
18		fvexd 6920	. . . . . . . . 9 ⊢ (𝑦 ∈ (Walks‘𝐺) → (2nd ‘𝑦) ∈ V)
19	9, 16, 17, 18	fvmptd3 7037	. . . . . . . 8 ⊢ (𝑦 ∈ (Walks‘𝐺) → (𝐹‘𝑦) = (2nd ‘𝑦))
20	15, 19	eqeqan12d 2777	. . . . . . 7 ⊢ ((𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (2nd ‘𝑥) = (2nd ‘𝑦)))
21	20	adantl 487	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (2nd ‘𝑥) = (2nd ‘𝑦)))
22		uspgr2wlkeqi 30255	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) → 𝑥 = 𝑦)
23	22	ad4ant134 1193	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) → 𝑥 = 𝑦)
24	23	ex 418	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((2nd ‘𝑥) = (2nd ‘𝑦) → 𝑥 = 𝑦))
25	21, 24	sylbid 243	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
26	25	ralrimivva 3209	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → ∀𝑥 ∈ (Walks‘𝐺)∀𝑦 ∈ (Walks‘𝐺)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
27		dff13 7278	. . . 4 ⊢ (𝐹:(Walks‘𝐺)⟶1-1(WWalks‘𝐺) ↔ (𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺) ∧ ∀𝑥 ∈ (Walks‘𝐺)∀𝑦 ∈ (Walks‘𝐺)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
28	11, 26, 27	sylanbrc 595	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)⟶1-1(WWalks‘𝐺))
29		wlkiswwlks 30492	. . . . . . . . . 10 ⊢ (𝐺 ∈ USPGraph → (∃𝑓 𝑓(Walks‘𝐺)𝑦 ↔ 𝑦 ∈ (WWalks‘𝐺)))
30	29	adantr 486	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → (∃𝑓 𝑓(Walks‘𝐺)𝑦 ↔ 𝑦 ∈ (WWalks‘𝐺)))
31		df-br 5107	. . . . . . . . . . 11 ⊢ (𝑓(Walks‘𝐺)𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺))
32		vex 3458	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
33		vex 3458	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
34	32, 33	op2nd 8031	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨𝑓, 𝑦⟩) = 𝑦
35	34	eqcomi 2772	. . . . . . . . . . . 12 ⊢ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)
36		opex 5436	. . . . . . . . . . . . 13 ⊢ ⟨𝑓, 𝑦⟩ ∈ V
37		eleq1 2852	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑥 ∈ (Walks‘𝐺) ↔ ⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺)))
38		fveq2 6905	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝑓, 𝑦⟩))
39	38	eqeq2d 2774	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑦 = (2nd ‘𝑥) ↔ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)))
40	37, 39	anbi12d 644	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → ((𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)) ↔ (⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩))))
41	36, 40	spcev 3564	. . . . . . . . . . . 12 ⊢ ((⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
42	35, 41	mpan2 704	. . . . . . . . . . 11 ⊢ (⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
43	31, 42	sylbi 220	. . . . . . . . . 10 ⊢ (𝑓(Walks‘𝐺)𝑦 → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
44	43	exlimiv 1965	. . . . . . . . 9 ⊢ (∃𝑓 𝑓(Walks‘𝐺)𝑦 → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
45	30, 44	biimtrrdi 257	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → (𝑦 ∈ (WWalks‘𝐺) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥))))
46	45	imp 412	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
47		df-rex 3091	. . . . . . 7 ⊢ (∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd ‘𝑥) ↔ ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
48	46, 47	sylibr 237	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd ‘𝑥))
49	15	eqeq2d 2774	. . . . . . 7 ⊢ (𝑥 ∈ (Walks‘𝐺) → (𝑦 = (𝐹‘𝑥) ↔ 𝑦 = (2nd ‘𝑥)))
50	49	rexbiia 3111	. . . . . 6 ⊢ (∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd ‘𝑥))
51	48, 50	sylibr 237	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹‘𝑥))
52	51	ralrimiva 3158	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → ∀𝑦 ∈ (WWalks‘𝐺)∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹‘𝑥))
53		dffo3 7123	. . . 4 ⊢ (𝐹:(Walks‘𝐺)⟶onto(WWalks‘𝐺) ↔ (𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺) ∧ ∀𝑦 ∈ (WWalks‘𝐺)∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹‘𝑥)))
54	11, 52, 53	sylanbrc 595	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)⟶onto(WWalks‘𝐺))
55		df-f1o 6562	. . 3 ⊢ (𝐹:(Walks‘𝐺)⟶1-1onto(WWalks‘𝐺) ↔ (𝐹:(Walks‘𝐺)⟶1-1(WWalks‘𝐺) ∧ 𝐹:(Walks‘𝐺)⟶onto(WWalks‘𝐺)))
56	28, 54, 55	sylanbrc 595	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)⟶1-1onto(WWalks‘𝐺))
57	10, 56	mpdan 700	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)⟶1-1onto(WWalks‘𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454  ⟨cop 4838   class class class wbr 5106   ↦ cmpt 5189  ⟶wf 6551  ⟶1-1wf1 6552  ⟶ontowfo 6553  ⟶1-1ontowf1o 6554  ‘cfv 6555  1^st c1st 8020  2^nd c2nd 8021  USPGraphcuspgr 29756  Walkscwlks 30204  WWalkscwwlks 30441

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-oadd 8505  df-er 8742  df-map 8874  df-pm 8875  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-dju 10006  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-n0 12631  df-xnn0 12704  df-z 12718  df-uz 12990  df-fz 13664  df-fzo 13813  df-hash 14499  df-word 14683  df-edg 29653  df-uhgr 29663  df-upgr 29687  df-uspgr 29758  df-wlks 30207  df-wwlks 30446

This theorem is used by:  wlkswwlksen  30496  wlknwwlksnbij  30504