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Theorem wlkswwlksf1o 27573
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 6679 . . . . . 6 (1st𝑤) ∈ V
2 breq1 5065 . . . . . 6 (𝑓 = (1st𝑤) → (𝑓(Walks‘𝐺)(2nd𝑤) ↔ (1st𝑤)(Walks‘𝐺)(2nd𝑤)))
31, 2spcev 3610 . . . . 5 ((1st𝑤)(Walks‘𝐺)(2nd𝑤) → ∃𝑓 𝑓(Walks‘𝐺)(2nd𝑤))
4 wlkiswwlks 27570 . . . . 5 (𝐺 ∈ USPGraph → (∃𝑓 𝑓(Walks‘𝐺)(2nd𝑤) ↔ (2nd𝑤) ∈ (WWalks‘𝐺)))
53, 4syl5ib 245 . . . 4 (𝐺 ∈ USPGraph → ((1st𝑤)(Walks‘𝐺)(2nd𝑤) → (2nd𝑤) ∈ (WWalks‘𝐺)))
6 wlkcpr 27326 . . . . 5 (𝑤 ∈ (Walks‘𝐺) ↔ (1st𝑤)(Walks‘𝐺)(2nd𝑤))
76biimpi 217 . . . 4 (𝑤 ∈ (Walks‘𝐺) → (1st𝑤)(Walks‘𝐺)(2nd𝑤))
85, 7impel 506 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (Walks‘𝐺)) → (2nd𝑤) ∈ (WWalks‘𝐺))
9 wlkswwlksf1o.f . . 3 𝐹 = (𝑤 ∈ (Walks‘𝐺) ↦ (2nd𝑤))
108, 9fmptd 6873 . 2 (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺))
11 simpr 485 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺))
12 fveq2 6666 . . . . . . . . 9 (𝑤 = 𝑥 → (2nd𝑤) = (2nd𝑥))
13 id 22 . . . . . . . . 9 (𝑥 ∈ (Walks‘𝐺) → 𝑥 ∈ (Walks‘𝐺))
14 fvexd 6681 . . . . . . . . 9 (𝑥 ∈ (Walks‘𝐺) → (2nd𝑥) ∈ V)
159, 12, 13, 14fvmptd3 6786 . . . . . . . 8 (𝑥 ∈ (Walks‘𝐺) → (𝐹𝑥) = (2nd𝑥))
16 fveq2 6666 . . . . . . . . 9 (𝑤 = 𝑦 → (2nd𝑤) = (2nd𝑦))
17 id 22 . . . . . . . . 9 (𝑦 ∈ (Walks‘𝐺) → 𝑦 ∈ (Walks‘𝐺))
18 fvexd 6681 . . . . . . . . 9 (𝑦 ∈ (Walks‘𝐺) → (2nd𝑦) ∈ V)
199, 16, 17, 18fvmptd3 6786 . . . . . . . 8 (𝑦 ∈ (Walks‘𝐺) → (𝐹𝑦) = (2nd𝑦))
2015, 19eqeqan12d 2842 . . . . . . 7 ((𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) → ((𝐹𝑥) = (𝐹𝑦) ↔ (2nd𝑥) = (2nd𝑦)))
2120adantl 482 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((𝐹𝑥) = (𝐹𝑦) ↔ (2nd𝑥) = (2nd𝑦)))
22 uspgr2wlkeqi 27345 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (2nd𝑥) = (2nd𝑦)) → 𝑥 = 𝑦)
2322ad4ant134 1168 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) ∧ (2nd𝑥) = (2nd𝑦)) → 𝑥 = 𝑦)
2423ex 413 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((2nd𝑥) = (2nd𝑦) → 𝑥 = 𝑦))
2521, 24sylbid 241 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
2625ralrimivva 3195 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → ∀𝑥 ∈ (Walks‘𝐺)∀𝑦 ∈ (Walks‘𝐺)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
27 dff13 7010 . . . 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 583 . . 3 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–1-1→(WWalks‘𝐺))
29 wlkiswwlks 27570 . . . . . . . . . 10 (𝐺 ∈ USPGraph → (∃𝑓 𝑓(Walks‘𝐺)𝑦𝑦 ∈ (WWalks‘𝐺)))
3029adantr 481 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → (∃𝑓 𝑓(Walks‘𝐺)𝑦𝑦 ∈ (WWalks‘𝐺)))
31 df-br 5063 . . . . . . . . . . 11 (𝑓(Walks‘𝐺)𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺))
32 vex 3502 . . . . . . . . . . . . . 14 𝑓 ∈ V
33 vex 3502 . . . . . . . . . . . . . 14 𝑦 ∈ V
3432, 33op2nd 7692 . . . . . . . . . . . . 13 (2nd ‘⟨𝑓, 𝑦⟩) = 𝑦
3534eqcomi 2834 . . . . . . . . . . . 12 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)
36 opex 5352 . . . . . . . . . . . . 13 𝑓, 𝑦⟩ ∈ V
37 eleq1 2904 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑥 ∈ (Walks‘𝐺) ↔ ⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺)))
38 fveq2 6666 . . . . . . . . . . . . . . 15 (𝑥 = ⟨𝑓, 𝑦⟩ → (2nd𝑥) = (2nd ‘⟨𝑓, 𝑦⟩))
3938eqeq2d 2836 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑦 = (2nd𝑥) ↔ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)))
4037, 39anbi12d 630 . . . . . . . . . . . . 13 (𝑥 = ⟨𝑓, 𝑦⟩ → ((𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)) ↔ (⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩))))
4136, 40spcev 3610 . . . . . . . . . . . 12 ((⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
4235, 41mpan2 687 . . . . . . . . . . 11 (⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
4331, 42sylbi 218 . . . . . . . . . 10 (𝑓(Walks‘𝐺)𝑦 → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
4443exlimiv 1924 . . . . . . . . 9 (∃𝑓 𝑓(Walks‘𝐺)𝑦 → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
4530, 44syl6bir 255 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → (𝑦 ∈ (WWalks‘𝐺) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥))))
4645imp 407 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
47 df-rex 3148 . . . . . . 7 (∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd𝑥) ↔ ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
4846, 47sylibr 235 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd𝑥))
4915eqeq2d 2836 . . . . . . 7 (𝑥 ∈ (Walks‘𝐺) → (𝑦 = (𝐹𝑥) ↔ 𝑦 = (2nd𝑥)))
5049rexbiia 3250 . . . . . 6 (∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹𝑥) ↔ ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd𝑥))
5148, 50sylibr 235 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹𝑥))
5251ralrimiva 3186 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → ∀𝑦 ∈ (WWalks‘𝐺)∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹𝑥))
53 dffo3 6863 . . . 4 (𝐹:(Walks‘𝐺)–onto→(WWalks‘𝐺) ↔ (𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺) ∧ ∀𝑦 ∈ (WWalks‘𝐺)∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹𝑥)))
5411, 52, 53sylanbrc 583 . . 3 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–onto→(WWalks‘𝐺))
55 df-f1o 6358 . . 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 583 . 2 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
5710, 56mpdan 683 1 (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  wral 3142  wrex 3143  Vcvv 3499  cop 4569   class class class wbr 5062  cmpt 5142  wf 6347  1-1wf1 6348  ontowfo 6349  1-1-ontowf1o 6350  cfv 6351  1st c1st 7681  2nd c2nd 7682  USPGraphcuspgr 26849  Walkscwlks 27294  WWalkscwwlks 27519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ifp 1057  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-map 8401  df-pm 8402  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12886  df-fzo 13027  df-hash 13684  df-word 13855  df-edg 26749  df-uhgr 26759  df-upgr 26783  df-uspgr 26851  df-wlks 27297  df-wwlks 27524
This theorem is referenced by:  wlkswwlksen  27574  wlknwwlksnbij  27582
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