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Theorem ispth 28980
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 28978 . . . 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 1096 . . . . 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 5216 . . . 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 2761 . . 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 486 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ 𝑝 = 𝑃)
6 fveq2 6892 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (β™―β€˜π‘“) = (β™―β€˜πΉ))
76oveq2d 7425 . . . . . . . 8 (𝑓 = 𝐹 β†’ (1..^(β™―β€˜π‘“)) = (1..^(β™―β€˜πΉ)))
87adantr 482 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (1..^(β™―β€˜π‘“)) = (1..^(β™―β€˜πΉ)))
95, 8reseq12d 5983 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (𝑝 β†Ύ (1..^(β™―β€˜π‘“))) = (𝑃 β†Ύ (1..^(β™―β€˜πΉ))))
109cnveqd 5876 . . . . 5 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ β—‘(𝑝 β†Ύ (1..^(β™―β€˜π‘“))) = β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))))
1110funeqd 6571 . . . 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 4745 . . . . . . . 8 (𝑓 = 𝐹 β†’ {0, (β™―β€˜π‘“)} = {0, (β™―β€˜πΉ)})
1312adantr 482 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ {0, (β™―β€˜π‘“)} = {0, (β™―β€˜πΉ)})
145, 13imaeq12d 6061 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (𝑝 β€œ {0, (β™―β€˜π‘“)}) = (𝑃 β€œ {0, (β™―β€˜πΉ)}))
155, 8imaeq12d 6061 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (𝑝 β€œ (1..^(β™―β€˜π‘“))) = (𝑃 β€œ (1..^(β™―β€˜πΉ))))
1614, 15ineq12d 4214 . . . . 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 2735 . . . 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 632 . . 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 28951 . . 3 Rel (Trailsβ€˜πΊ)
204, 18, 19brfvopabrbr 6996 . 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 1096 . 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 278 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 397   ∧ w3a 1088   = wceq 1542   ∩ cin 3948  βˆ…c0 4323  {cpr 4631   class class class wbr 5149  {copab 5211  β—‘ccnv 5676   β†Ύ cres 5679   β€œ cima 5680  Fun wfun 6538  β€˜cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111  ..^cfzo 13627  β™―chash 14290  Trailsctrls 28947  Pathscpths 28969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-trls 28949  df-pths 28973
This theorem is referenced by:  pthistrl  28982  spthispth  28983  pthdivtx  28986  2pthnloop  28988  pthdepisspth  28992  pthd  29026  0pth  29378  1pthd  29396  pthhashvtx  34118  subgrpth  34125
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