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Theorem wlkswwlksf1o 27764
 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) ↦ (2nd𝑤))
Assertion
Ref Expression
wlkswwlksf1o (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)–1-1-onto→(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.
StepHypRef Expression
1 fvex 6671 . . . . . 6 (1st𝑤) ∈ V
2 breq1 5035 . . . . . 6 (𝑓 = (1st𝑤) → (𝑓(Walks‘𝐺)(2nd𝑤) ↔ (1st𝑤)(Walks‘𝐺)(2nd𝑤)))
31, 2spcev 3525 . . . . 5 ((1st𝑤)(Walks‘𝐺)(2nd𝑤) → ∃𝑓 𝑓(Walks‘𝐺)(2nd𝑤))
4 wlkiswwlks 27761 . . . . 5 (𝐺 ∈ USPGraph → (∃𝑓 𝑓(Walks‘𝐺)(2nd𝑤) ↔ (2nd𝑤) ∈ (WWalks‘𝐺)))
53, 4syl5ib 247 . . . 4 (𝐺 ∈ USPGraph → ((1st𝑤)(Walks‘𝐺)(2nd𝑤) → (2nd𝑤) ∈ (WWalks‘𝐺)))
6 wlkcpr 27517 . . . . 5 (𝑤 ∈ (Walks‘𝐺) ↔ (1st𝑤)(Walks‘𝐺)(2nd𝑤))
76biimpi 219 . . . 4 (𝑤 ∈ (Walks‘𝐺) → (1st𝑤)(Walks‘𝐺)(2nd𝑤))
85, 7impel 509 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (Walks‘𝐺)) → (2nd𝑤) ∈ (WWalks‘𝐺))
9 wlkswwlksf1o.f . . 3 𝐹 = (𝑤 ∈ (Walks‘𝐺) ↦ (2nd𝑤))
108, 9fmptd 6869 . 2 (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺))
11 simpr 488 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺))
12 fveq2 6658 . . . . . . . . 9 (𝑤 = 𝑥 → (2nd𝑤) = (2nd𝑥))
13 id 22 . . . . . . . . 9 (𝑥 ∈ (Walks‘𝐺) → 𝑥 ∈ (Walks‘𝐺))
14 fvexd 6673 . . . . . . . . 9 (𝑥 ∈ (Walks‘𝐺) → (2nd𝑥) ∈ V)
159, 12, 13, 14fvmptd3 6782 . . . . . . . 8 (𝑥 ∈ (Walks‘𝐺) → (𝐹𝑥) = (2nd𝑥))
16 fveq2 6658 . . . . . . . . 9 (𝑤 = 𝑦 → (2nd𝑤) = (2nd𝑦))
17 id 22 . . . . . . . . 9 (𝑦 ∈ (Walks‘𝐺) → 𝑦 ∈ (Walks‘𝐺))
18 fvexd 6673 . . . . . . . . 9 (𝑦 ∈ (Walks‘𝐺) → (2nd𝑦) ∈ V)
199, 16, 17, 18fvmptd3 6782 . . . . . . . 8 (𝑦 ∈ (Walks‘𝐺) → (𝐹𝑦) = (2nd𝑦))
2015, 19eqeqan12d 2775 . . . . . . 7 ((𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) → ((𝐹𝑥) = (𝐹𝑦) ↔ (2nd𝑥) = (2nd𝑦)))
2120adantl 485 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((𝐹𝑥) = (𝐹𝑦) ↔ (2nd𝑥) = (2nd𝑦)))
22 uspgr2wlkeqi 27536 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (2nd𝑥) = (2nd𝑦)) → 𝑥 = 𝑦)
2322ad4ant134 1171 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) ∧ (2nd𝑥) = (2nd𝑦)) → 𝑥 = 𝑦)
2423ex 416 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((2nd𝑥) = (2nd𝑦) → 𝑥 = 𝑦))
2521, 24sylbid 243 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
2625ralrimivva 3120 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → ∀𝑥 ∈ (Walks‘𝐺)∀𝑦 ∈ (Walks‘𝐺)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
27 dff13 7005 . . . 4 (𝐹:(Walks‘𝐺)–1-1→(WWalks‘𝐺) ↔ (𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺) ∧ ∀𝑥 ∈ (Walks‘𝐺)∀𝑦 ∈ (Walks‘𝐺)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
2811, 26, 27sylanbrc 586 . . 3 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–1-1→(WWalks‘𝐺))
29 wlkiswwlks 27761 . . . . . . . . . 10 (𝐺 ∈ USPGraph → (∃𝑓 𝑓(Walks‘𝐺)𝑦𝑦 ∈ (WWalks‘𝐺)))
3029adantr 484 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → (∃𝑓 𝑓(Walks‘𝐺)𝑦𝑦 ∈ (WWalks‘𝐺)))
31 df-br 5033 . . . . . . . . . . 11 (𝑓(Walks‘𝐺)𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺))
32 vex 3413 . . . . . . . . . . . . . 14 𝑓 ∈ V
33 vex 3413 . . . . . . . . . . . . . 14 𝑦 ∈ V
3432, 33op2nd 7702 . . . . . . . . . . . . 13 (2nd ‘⟨𝑓, 𝑦⟩) = 𝑦
3534eqcomi 2767 . . . . . . . . . . . 12 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)
36 opex 5324 . . . . . . . . . . . . 13 𝑓, 𝑦⟩ ∈ V
37 eleq1 2839 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑥 ∈ (Walks‘𝐺) ↔ ⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺)))
38 fveq2 6658 . . . . . . . . . . . . . . 15 (𝑥 = ⟨𝑓, 𝑦⟩ → (2nd𝑥) = (2nd ‘⟨𝑓, 𝑦⟩))
3938eqeq2d 2769 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑦 = (2nd𝑥) ↔ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)))
4037, 39anbi12d 633 . . . . . . . . . . . . 13 (𝑥 = ⟨𝑓, 𝑦⟩ → ((𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)) ↔ (⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩))))
4136, 40spcev 3525 . . . . . . . . . . . 12 ((⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
4235, 41mpan2 690 . . . . . . . . . . 11 (⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
4331, 42sylbi 220 . . . . . . . . . 10 (𝑓(Walks‘𝐺)𝑦 → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
4443exlimiv 1931 . . . . . . . . 9 (∃𝑓 𝑓(Walks‘𝐺)𝑦 → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
4530, 44syl6bir 257 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → (𝑦 ∈ (WWalks‘𝐺) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥))))
4645imp 410 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
47 df-rex 3076 . . . . . . 7 (∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd𝑥) ↔ ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
4846, 47sylibr 237 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd𝑥))
4915eqeq2d 2769 . . . . . . 7 (𝑥 ∈ (Walks‘𝐺) → (𝑦 = (𝐹𝑥) ↔ 𝑦 = (2nd𝑥)))
5049rexbiia 3174 . . . . . 6 (∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹𝑥) ↔ ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd𝑥))
5148, 50sylibr 237 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹𝑥))
5251ralrimiva 3113 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → ∀𝑦 ∈ (WWalks‘𝐺)∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹𝑥))
53 dffo3 6859 . . . 4 (𝐹:(Walks‘𝐺)–onto→(WWalks‘𝐺) ↔ (𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺) ∧ ∀𝑦 ∈ (WWalks‘𝐺)∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹𝑥)))
5411, 52, 53sylanbrc 586 . . 3 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–onto→(WWalks‘𝐺))
55 df-f1o 6342 . . 3 (𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺) ↔ (𝐹:(Walks‘𝐺)–1-1→(WWalks‘𝐺) ∧ 𝐹:(Walks‘𝐺)–onto→(WWalks‘𝐺)))
5628, 54, 55sylanbrc 586 . 2 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
5710, 56mpdan 686 1 (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  Vcvv 3409  ⟨cop 4528   class class class wbr 5032   ↦ cmpt 5112  ⟶wf 6331  –1-1→wf1 6332  –onto→wfo 6333  –1-1-onto→wf1o 6334  ‘cfv 6335  1st c1st 7691  2nd c2nd 7692  USPGraphcuspgr 27040  Walkscwlks 27485  WWalkscwwlks 27710 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-2o 8113  df-oadd 8116  df-er 8299  df-map 8418  df-pm 8419  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-dju 9363  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-n0 11935  df-xnn0 12007  df-z 12021  df-uz 12283  df-fz 12940  df-fzo 13083  df-hash 13741  df-word 13914  df-edg 26940  df-uhgr 26950  df-upgr 26974  df-uspgr 27042  df-wlks 27488  df-wwlks 27715 This theorem is referenced by:  wlkswwlksen  27765  wlknwwlksnbij  27773
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