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Theorem ispth 26098
 Description: Properties of a pair of functions to be a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
Assertion
Ref Expression
ispth (((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 Paths 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)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 df-br 4584 . . 3 (𝐹(𝑉 Paths 𝐸)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (𝑉 Paths 𝐸))
2 pths 26096 . . . . 5 ((𝑉𝑋𝐸𝑌) → (𝑉 Paths 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
32adantr 480 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝑉 Paths 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
43eleq2d 2673 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (⟨𝐹, 𝑃⟩ ∈ (𝑉 Paths 𝐸) ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)}))
51, 4syl5bb 271 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 Paths 𝐸)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)}))
6 breq12 4588 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓(𝑉 Trails 𝐸)𝑝𝐹(𝑉 Trails 𝐸)𝑃))
7 simpr 476 . . . . . . . 8 ((𝑓 = 𝐹𝑝 = 𝑃) → 𝑝 = 𝑃)
8 fveq2 6103 . . . . . . . . . 10 (𝑓 = 𝐹 → (#‘𝑓) = (#‘𝐹))
98adantr 480 . . . . . . . . 9 ((𝑓 = 𝐹𝑝 = 𝑃) → (#‘𝑓) = (#‘𝐹))
109oveq2d 6565 . . . . . . . 8 ((𝑓 = 𝐹𝑝 = 𝑃) → (1..^(#‘𝑓)) = (1..^(#‘𝐹)))
117, 10reseq12d 5318 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝 ↾ (1..^(#‘𝑓))) = (𝑃 ↾ (1..^(#‘𝐹))))
1211cnveqd 5220 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝 ↾ (1..^(#‘𝑓))) = (𝑃 ↾ (1..^(#‘𝐹))))
1312funeqd 5825 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (Fun (𝑝 ↾ (1..^(#‘𝑓))) ↔ Fun (𝑃 ↾ (1..^(#‘𝐹)))))
148preq2d 4219 . . . . . . . . 9 (𝑓 = 𝐹 → {0, (#‘𝑓)} = {0, (#‘𝐹)})
1514adantr 480 . . . . . . . 8 ((𝑓 = 𝐹𝑝 = 𝑃) → {0, (#‘𝑓)} = {0, (#‘𝐹)})
167, 15imaeq12d 5386 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝 “ {0, (#‘𝑓)}) = (𝑃 “ {0, (#‘𝐹)}))
178oveq2d 6565 . . . . . . . . 9 (𝑓 = 𝐹 → (1..^(#‘𝑓)) = (1..^(#‘𝐹)))
1817adantr 480 . . . . . . . 8 ((𝑓 = 𝐹𝑝 = 𝑃) → (1..^(#‘𝑓)) = (1..^(#‘𝐹)))
197, 18imaeq12d 5386 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝 “ (1..^(#‘𝑓))) = (𝑃 “ (1..^(#‘𝐹))))
2016, 19ineq12d 3777 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))))
2120eqeq1d 2612 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅ ↔ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅))
226, 13, 213anbi123d 1391 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅) ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun (𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)))
2322opelopabga 4913 . . 3 ((𝐹𝑊𝑃𝑍) → (⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)} ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun (𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)))
2423adantl 481 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)} ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun (𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)))
255, 24bitrd 267 1 (((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 Paths 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun (𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∩ cin 3539  ∅c0 3874  {cpr 4127  ⟨cop 4131   class class class wbr 4583  {copab 4642  ◡ccnv 5037   ↾ cres 5040   “ cima 5041  Fun wfun 5798  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  ..^cfzo 12334  #chash 12979   Trails ctrail 26027   Paths cpath 26028 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-wlk 26036  df-trail 26037  df-pth 26038 This theorem is referenced by:  0pth  26100  pthistrl  26102  spthispth  26103  pthdepisspth  26104  1pthon  26121  constr2pth  26131  3v3e3cycl1  26172  constr3pth  26188  4cycl4v4e  26194  4cycl4dv4e  26196
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