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Theorem wlkpwwlkf1ouspgr 26833
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
wlkpwwlkf1ouspgr.f 𝐹 = (𝑤 ∈ (Walks‘𝐺) ↦ (2nd𝑤))
Assertion
Ref Expression
wlkpwwlkf1ouspgr (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
Distinct variable group:   𝑤,𝐺
Allowed substitution hint:   𝐹(𝑤)

Proof of Theorem wlkpwwlkf1ouspgr
Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6239 . . . . . 6 (1st𝑤) ∈ V
2 breq1 4688 . . . . . 6 (𝑓 = (1st𝑤) → (𝑓(Walks‘𝐺)(2nd𝑤) ↔ (1st𝑤)(Walks‘𝐺)(2nd𝑤)))
31, 2spcev 3331 . . . . 5 ((1st𝑤)(Walks‘𝐺)(2nd𝑤) → ∃𝑓 𝑓(Walks‘𝐺)(2nd𝑤))
4 wlkiswwlks 26830 . . . . 5 (𝐺 ∈ USPGraph → (∃𝑓 𝑓(Walks‘𝐺)(2nd𝑤) ↔ (2nd𝑤) ∈ (WWalks‘𝐺)))
53, 4syl5ib 234 . . . 4 (𝐺 ∈ USPGraph → ((1st𝑤)(Walks‘𝐺)(2nd𝑤) → (2nd𝑤) ∈ (WWalks‘𝐺)))
6 wlkcpr 26580 . . . . 5 (𝑤 ∈ (Walks‘𝐺) ↔ (1st𝑤)(Walks‘𝐺)(2nd𝑤))
76biimpi 206 . . . 4 (𝑤 ∈ (Walks‘𝐺) → (1st𝑤)(Walks‘𝐺)(2nd𝑤))
85, 7impel 484 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (Walks‘𝐺)) → (2nd𝑤) ∈ (WWalks‘𝐺))
9 wlkpwwlkf1ouspgr.f . . 3 𝐹 = (𝑤 ∈ (Walks‘𝐺) ↦ (2nd𝑤))
108, 9fmptd 6425 . 2 (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺))
11 simpr 476 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺))
129a1i 11 . . . . . . . . 9 (𝑥 ∈ (Walks‘𝐺) → 𝐹 = (𝑤 ∈ (Walks‘𝐺) ↦ (2nd𝑤)))
13 fveq2 6229 . . . . . . . . . 10 (𝑤 = 𝑥 → (2nd𝑤) = (2nd𝑥))
1413adantl 481 . . . . . . . . 9 ((𝑥 ∈ (Walks‘𝐺) ∧ 𝑤 = 𝑥) → (2nd𝑤) = (2nd𝑥))
15 id 22 . . . . . . . . 9 (𝑥 ∈ (Walks‘𝐺) → 𝑥 ∈ (Walks‘𝐺))
16 fvexd 6241 . . . . . . . . 9 (𝑥 ∈ (Walks‘𝐺) → (2nd𝑥) ∈ V)
1712, 14, 15, 16fvmptd 6327 . . . . . . . 8 (𝑥 ∈ (Walks‘𝐺) → (𝐹𝑥) = (2nd𝑥))
189a1i 11 . . . . . . . . 9 (𝑦 ∈ (Walks‘𝐺) → 𝐹 = (𝑤 ∈ (Walks‘𝐺) ↦ (2nd𝑤)))
19 fveq2 6229 . . . . . . . . . 10 (𝑤 = 𝑦 → (2nd𝑤) = (2nd𝑦))
2019adantl 481 . . . . . . . . 9 ((𝑦 ∈ (Walks‘𝐺) ∧ 𝑤 = 𝑦) → (2nd𝑤) = (2nd𝑦))
21 id 22 . . . . . . . . 9 (𝑦 ∈ (Walks‘𝐺) → 𝑦 ∈ (Walks‘𝐺))
22 fvexd 6241 . . . . . . . . 9 (𝑦 ∈ (Walks‘𝐺) → (2nd𝑦) ∈ V)
2318, 20, 21, 22fvmptd 6327 . . . . . . . 8 (𝑦 ∈ (Walks‘𝐺) → (𝐹𝑦) = (2nd𝑦))
2417, 23eqeqan12d 2667 . . . . . . 7 ((𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) → ((𝐹𝑥) = (𝐹𝑦) ↔ (2nd𝑥) = (2nd𝑦)))
2524adantl 481 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((𝐹𝑥) = (𝐹𝑦) ↔ (2nd𝑥) = (2nd𝑦)))
26 uspgr2wlkeqi 26600 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (2nd𝑥) = (2nd𝑦)) → 𝑥 = 𝑦)
2726ad4ant134 1323 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) ∧ (2nd𝑥) = (2nd𝑦)) → 𝑥 = 𝑦)
2827ex 449 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((2nd𝑥) = (2nd𝑦) → 𝑥 = 𝑦))
2925, 28sylbid 230 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3029ralrimivva 3000 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → ∀𝑥 ∈ (Walks‘𝐺)∀𝑦 ∈ (Walks‘𝐺)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
31 dff13 6552 . . . 4 (𝐹:(Walks‘𝐺)–1-1→(WWalks‘𝐺) ↔ (𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺) ∧ ∀𝑥 ∈ (Walks‘𝐺)∀𝑦 ∈ (Walks‘𝐺)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3211, 30, 31sylanbrc 699 . . 3 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–1-1→(WWalks‘𝐺))
33 wlkiswwlks 26830 . . . . . . . . . 10 (𝐺 ∈ USPGraph → (∃𝑓 𝑓(Walks‘𝐺)𝑦𝑦 ∈ (WWalks‘𝐺)))
3433adantr 480 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → (∃𝑓 𝑓(Walks‘𝐺)𝑦𝑦 ∈ (WWalks‘𝐺)))
35 df-br 4686 . . . . . . . . . . 11 (𝑓(Walks‘𝐺)𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺))
36 vex 3234 . . . . . . . . . . . . . 14 𝑓 ∈ V
37 vex 3234 . . . . . . . . . . . . . 14 𝑦 ∈ V
3836, 37op2nd 7219 . . . . . . . . . . . . 13 (2nd ‘⟨𝑓, 𝑦⟩) = 𝑦
3938eqcomi 2660 . . . . . . . . . . . 12 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)
40 opex 4962 . . . . . . . . . . . . 13 𝑓, 𝑦⟩ ∈ V
41 eleq1 2718 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑥 ∈ (Walks‘𝐺) ↔ ⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺)))
42 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑥 = ⟨𝑓, 𝑦⟩ → (2nd𝑥) = (2nd ‘⟨𝑓, 𝑦⟩))
4342eqeq2d 2661 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑦 = (2nd𝑥) ↔ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)))
4441, 43anbi12d 747 . . . . . . . . . . . . 13 (𝑥 = ⟨𝑓, 𝑦⟩ → ((𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)) ↔ (⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩))))
4540, 44spcev 3331 . . . . . . . . . . . 12 ((⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
4639, 45mpan2 707 . . . . . . . . . . 11 (⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
4735, 46sylbi 207 . . . . . . . . . 10 (𝑓(Walks‘𝐺)𝑦 → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
4847exlimiv 1898 . . . . . . . . 9 (∃𝑓 𝑓(Walks‘𝐺)𝑦 → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
4934, 48syl6bir 244 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → (𝑦 ∈ (WWalks‘𝐺) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥))))
5049imp 444 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
51 df-rex 2947 . . . . . . 7 (∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd𝑥) ↔ ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
5250, 51sylibr 224 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd𝑥))
5317eqeq2d 2661 . . . . . . 7 (𝑥 ∈ (Walks‘𝐺) → (𝑦 = (𝐹𝑥) ↔ 𝑦 = (2nd𝑥)))
5453rexbiia 3069 . . . . . 6 (∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹𝑥) ↔ ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd𝑥))
5552, 54sylibr 224 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹𝑥))
5655ralrimiva 2995 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → ∀𝑦 ∈ (WWalks‘𝐺)∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹𝑥))
57 dffo3 6414 . . . 4 (𝐹:(Walks‘𝐺)–onto→(WWalks‘𝐺) ↔ (𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺) ∧ ∀𝑦 ∈ (WWalks‘𝐺)∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹𝑥)))
5811, 56, 57sylanbrc 699 . . 3 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–onto→(WWalks‘𝐺))
59 df-f1o 5933 . . 3 (𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺) ↔ (𝐹:(Walks‘𝐺)–1-1→(WWalks‘𝐺) ∧ 𝐹:(Walks‘𝐺)–onto→(WWalks‘𝐺)))
6032, 58, 59sylanbrc 699 . 2 ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
6110, 60mpdan 703 1 (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cop 4216   class class class wbr 4685  cmpt 4762  wf 5922  1-1wf1 5923  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926  1st c1st 7208  2nd c2nd 7209  USPGraphcuspgr 26088  Walkscwlks 26548  WWalkscwwlks 26773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-edg 25985  df-uhgr 25998  df-upgr 26022  df-uspgr 26090  df-wlks 26551  df-wwlks 26778
This theorem is referenced by:  wlkisowwlkupgr  26834
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