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Theorem ispth 26675
Description: Conditions for a pair of classes/functions to be a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.)
Assertion
Ref Expression
ispth (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun (𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅))

Proof of Theorem ispth
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pthsfval 26673 . . . 4 (Paths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)}
2 3anass 1059 . . . . 5 ((𝑓(Trails‘𝐺)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅) ↔ (𝑓(Trails‘𝐺)𝑝 ∧ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)))
32opabbii 4750 . . . 4 {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅))}
41, 3eqtri 2673 . . 3 (Paths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅))}
5 simpr 476 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → 𝑝 = 𝑃)
6 fveq2 6229 . . . . . . . . 9 (𝑓 = 𝐹 → (#‘𝑓) = (#‘𝐹))
76oveq2d 6706 . . . . . . . 8 (𝑓 = 𝐹 → (1..^(#‘𝑓)) = (1..^(#‘𝐹)))
87adantr 480 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → (1..^(#‘𝑓)) = (1..^(#‘𝐹)))
95, 8reseq12d 5429 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝 ↾ (1..^(#‘𝑓))) = (𝑃 ↾ (1..^(#‘𝐹))))
109cnveqd 5330 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝 ↾ (1..^(#‘𝑓))) = (𝑃 ↾ (1..^(#‘𝐹))))
1110funeqd 5948 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (Fun (𝑝 ↾ (1..^(#‘𝑓))) ↔ Fun (𝑃 ↾ (1..^(#‘𝐹)))))
126preq2d 4307 . . . . . . . 8 (𝑓 = 𝐹 → {0, (#‘𝑓)} = {0, (#‘𝐹)})
1312adantr 480 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → {0, (#‘𝑓)} = {0, (#‘𝐹)})
145, 13imaeq12d 5502 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝 “ {0, (#‘𝑓)}) = (𝑃 “ {0, (#‘𝐹)}))
155, 8imaeq12d 5502 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝 “ (1..^(#‘𝑓))) = (𝑃 “ (1..^(#‘𝐹))))
1614, 15ineq12d 3848 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))))
1716eqeq1d 2653 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅ ↔ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅))
1811, 17anbi12d 747 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → ((Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅) ↔ (Fun (𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)))
19 reltrls 26647 . . 3 Rel (Trails‘𝐺)
204, 18, 19brfvopabrbr 6318 . 2 (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ (Fun (𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)))
21 3anass 1059 . 2 ((𝐹(Trails‘𝐺)𝑃 ∧ Fun (𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) ↔ (𝐹(Trails‘𝐺)𝑃 ∧ (Fun (𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)))
2220, 21bitr4i 267 1 (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun (𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1054   = wceq 1523  cin 3606  c0 3948  {cpr 4212   class class class wbr 4685  {copab 4745  ccnv 5142  cres 5145  cima 5146  Fun wfun 5920  cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975  ..^cfzo 12504  #chash 13157  Trailsctrls 26643  Pathscpths 26664
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-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-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  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-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-wlks 26551  df-trls 26645  df-pths 26668
This theorem is referenced by:  pthistrl  26677  spthispth  26678  pthdivtx  26681  2pthnloop  26683  pthdepisspth  26687  pthd  26721  0pth  27103  1pthd  27121
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