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Theorem ispth 29741
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 29739 . . . 4 (Paths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun (𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)}
2 3anass 1095 . . . . 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 5210 . . . 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 2765 . . 3 (Paths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ (Fun (𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅))}
5 simpr 484 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → 𝑝 = 𝑃)
6 fveq2 6906 . . . . . . . . 9 (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
76oveq2d 7447 . . . . . . . 8 (𝑓 = 𝐹 → (1..^(♯‘𝑓)) = (1..^(♯‘𝐹)))
87adantr 480 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → (1..^(♯‘𝑓)) = (1..^(♯‘𝐹)))
95, 8reseq12d 5998 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝 ↾ (1..^(♯‘𝑓))) = (𝑃 ↾ (1..^(♯‘𝐹))))
109cnveqd 5886 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝 ↾ (1..^(♯‘𝑓))) = (𝑃 ↾ (1..^(♯‘𝐹))))
1110funeqd 6588 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (Fun (𝑝 ↾ (1..^(♯‘𝑓))) ↔ Fun (𝑃 ↾ (1..^(♯‘𝐹)))))
126preq2d 4740 . . . . . . . 8 (𝑓 = 𝐹 → {0, (♯‘𝑓)} = {0, (♯‘𝐹)})
1312adantr 480 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → {0, (♯‘𝑓)} = {0, (♯‘𝐹)})
145, 13imaeq12d 6079 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝 “ {0, (♯‘𝑓)}) = (𝑃 “ {0, (♯‘𝐹)}))
155, 8imaeq12d 6079 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝 “ (1..^(♯‘𝑓))) = (𝑃 “ (1..^(♯‘𝐹))))
1614, 15ineq12d 4221 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))))
1716eqeq1d 2739 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅ ↔ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
1811, 17anbi12d 632 . . 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 29712 . . 3 Rel (Trails‘𝐺)
204, 18, 19brfvopabrbr 7013 . 2 (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ (Fun (𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅)))
21 3anass 1095 . 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 278 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 206  wa 395  w3a 1087   = wceq 1540  cin 3950  c0 4333  {cpr 4628   class class class wbr 5143  {copab 5205  ccnv 5684  cres 5687  cima 5688  Fun wfun 6555  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156  ..^cfzo 13694  chash 14369  Trailsctrls 29708  Pathscpths 29730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-trls 29710  df-pths 29734
This theorem is referenced by:  pthistrl  29743  spthispth  29744  pthdivtx  29747  dfpth2  29749  2pthnloop  29751  pthdepisspth  29755  pthd  29789  0pth  30144  1pthd  30162  pthhashvtx  35133  subgrpth  35139
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