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Theorem upgrwlkdvspth 26691
 Description: A walk consisting of different vertices is a simple path. Notice that this theorem would not hold for arbitrary hypergraphs, see the counterexample given in the comment of upgrspthswlk 26690. (Contributed by Alexander van der Vekens, 27-Oct-2017.) (Revised by AV, 17-Jan-2021.)
Assertion
Ref Expression
upgrwlkdvspth ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → 𝐹(SPaths‘𝐺)𝑃)

Proof of Theorem upgrwlkdvspth
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpc 1080 . 2 ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃))
2 upgrspthswlk 26690 . . . . 5 (𝐺 ∈ UPGraph → (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)})
323ad2ant1 1102 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)})
43breqd 4696 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → (𝐹(SPaths‘𝐺)𝑃𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)}𝑃))
5 wlkv 26564 . . . . . 6 (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
6 3simpc 1080 . . . . . 6 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹 ∈ V ∧ 𝑃 ∈ V))
75, 6syl 17 . . . . 5 (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ V ∧ 𝑃 ∈ V))
873ad2ant2 1103 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → (𝐹 ∈ V ∧ 𝑃 ∈ V))
9 breq12 4690 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓(Walks‘𝐺)𝑝𝐹(Walks‘𝐺)𝑃))
10 cnveq 5328 . . . . . . . 8 (𝑝 = 𝑃𝑝 = 𝑃)
1110funeqd 5948 . . . . . . 7 (𝑝 = 𝑃 → (Fun 𝑝 ↔ Fun 𝑃))
1211adantl 481 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (Fun 𝑝 ↔ Fun 𝑃))
139, 12anbi12d 747 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝) ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃)))
14 eqid 2651 . . . . 5 {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)}
1513, 14brabga 5018 . . . 4 ((𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)}𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃)))
168, 15syl 17 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → (𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)}𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃)))
174, 16bitrd 268 . 2 ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃)))
181, 17mpbird 247 1 ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → 𝐹(SPaths‘𝐺)𝑃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  Vcvv 3231   class class class wbr 4685  {copab 4745  ◡ccnv 5142  Fun wfun 5920  ‘cfv 5926  UPGraphcupgr 26020  Walkscwlks 26548  SPathscspths 26665 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-wlks 26551  df-trls 26645  df-spths 26669 This theorem is referenced by: (None)
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