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Theorem ispth 28977
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(Pathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…))
Proof of Theorem ispth
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pthsfval 28975 . . . 4 (Pathsβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Trailsβ€˜πΊ)𝑝 ∧ Fun β—‘(𝑝 β†Ύ (1..^(β™―β€˜π‘“))) ∧ ((𝑝 β€œ {0, (β™―β€˜π‘“)}) ∩ (𝑝 β€œ (1..^(β™―β€˜π‘“)))) = βˆ…)}
2 3anass 1095 . . . . 5 ((𝑓(Trailsβ€˜πΊ)𝑝 ∧ Fun β—‘(𝑝 β†Ύ (1..^(β™―β€˜π‘“))) ∧ ((𝑝 β€œ {0, (β™―β€˜π‘“)}) ∩ (𝑝 β€œ (1..^(β™―β€˜π‘“)))) = βˆ…) ↔ (𝑓(Trailsβ€˜πΊ)𝑝 ∧ (Fun β—‘(𝑝 β†Ύ (1..^(β™―β€˜π‘“))) ∧ ((𝑝 β€œ {0, (β™―β€˜π‘“)}) ∩ (𝑝 β€œ (1..^(β™―β€˜π‘“)))) = βˆ…)))
32opabbii 5215 . . . 4 {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Trailsβ€˜πΊ)𝑝 ∧ Fun β—‘(𝑝 β†Ύ (1..^(β™―β€˜π‘“))) ∧ ((𝑝 β€œ {0, (β™―β€˜π‘“)}) ∩ (𝑝 β€œ (1..^(β™―β€˜π‘“)))) = βˆ…)} = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Trailsβ€˜πΊ)𝑝 ∧ (Fun β—‘(𝑝 β†Ύ (1..^(β™―β€˜π‘“))) ∧ ((𝑝 β€œ {0, (β™―β€˜π‘“)}) ∩ (𝑝 β€œ (1..^(β™―β€˜π‘“)))) = βˆ…))}
41, 3eqtri 2760 . . 3 (Pathsβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Trailsβ€˜πΊ)𝑝 ∧ (Fun β—‘(𝑝 β†Ύ (1..^(β™―β€˜π‘“))) ∧ ((𝑝 β€œ {0, (β™―β€˜π‘“)}) ∩ (𝑝 β€œ (1..^(β™―β€˜π‘“)))) = βˆ…))}
5 simpr 485 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ 𝑝 = 𝑃)
6 fveq2 6891 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (β™―β€˜π‘“) = (β™―β€˜πΉ))
76oveq2d 7424 . . . . . . . 8 (𝑓 = 𝐹 β†’ (1..^(β™―β€˜π‘“)) = (1..^(β™―β€˜πΉ)))
87adantr 481 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (1..^(β™―β€˜π‘“)) = (1..^(β™―β€˜πΉ)))
95, 8reseq12d 5982 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (𝑝 β†Ύ (1..^(β™―β€˜π‘“))) = (𝑃 β†Ύ (1..^(β™―β€˜πΉ))))
109cnveqd 5875 . . . . 5 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ β—‘(𝑝 β†Ύ (1..^(β™―β€˜π‘“))) = β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))))
1110funeqd 6570 . . . 4 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (Fun β—‘(𝑝 β†Ύ (1..^(β™―β€˜π‘“))) ↔ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ)))))
126preq2d 4744 . . . . . . . 8 (𝑓 = 𝐹 β†’ {0, (β™―β€˜π‘“)} = {0, (β™―β€˜πΉ)})
1312adantr 481 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ {0, (β™―β€˜π‘“)} = {0, (β™―β€˜πΉ)})
145, 13imaeq12d 6060 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (𝑝 β€œ {0, (β™―β€˜π‘“)}) = (𝑃 β€œ {0, (β™―β€˜πΉ)}))
155, 8imaeq12d 6060 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (𝑝 β€œ (1..^(β™―β€˜π‘“))) = (𝑃 β€œ (1..^(β™―β€˜πΉ))))
1614, 15ineq12d 4213 . . . . 5 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ ((𝑝 β€œ {0, (β™―β€˜π‘“)}) ∩ (𝑝 β€œ (1..^(β™―β€˜π‘“)))) = ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))))
1716eqeq1d 2734 . . . 4 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (((𝑝 β€œ {0, (β™―β€˜π‘“)}) ∩ (𝑝 β€œ (1..^(β™―β€˜π‘“)))) = βˆ… ↔ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…))
1811, 17anbi12d 631 . . 3 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ ((Fun β—‘(𝑝 β†Ύ (1..^(β™―β€˜π‘“))) ∧ ((𝑝 β€œ {0, (β™―β€˜π‘“)}) ∩ (𝑝 β€œ (1..^(β™―β€˜π‘“)))) = βˆ…) ↔ (Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…)))
19 reltrls 28948 . . 3 Rel (Trailsβ€˜πΊ)
204, 18, 19brfvopabrbr 6995 . 2 (𝐹(Pathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ (Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…)))
21 3anass 1095 . 2 ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ (Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…)))
2220, 21bitr4i 277 1 (𝐹(Pathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∩ cin 3947  βˆ…c0 4322  {cpr 4630   class class class wbr 5148  {copab 5210  β—‘ccnv 5675   β†Ύ cres 5678   β€œ cima 5679  Fun wfun 6537  β€˜cfv 6543  (class class class)co 7408  0cc0 11109  1c1 11110  ..^cfzo 13626  β™―chash 14289  Trailsctrls 28944  Pathscpths 28966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-trls 28946  df-pths 28970
This theorem is referenced by:  pthistrl  28979  spthispth  28980  pthdivtx  28983  2pthnloop  28985  pthdepisspth  28989  pthd  29023  0pth  29375  1pthd  29393  pthhashvtx  34113  subgrpth  34120
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