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Theorem eupths 30023
Description: The Eulerian paths on the graph 𝐺. (Contributed by AV, 18-Feb-2021.) (Revised by AV, 29-Oct-2021.)
Hypothesis
Ref Expression
eupths.i 𝐼 = (iEdgβ€˜πΊ)
Assertion
Ref Expression
eupths (EulerPathsβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Trailsβ€˜πΊ)𝑝 ∧ 𝑓:(0..^(β™―β€˜π‘“))–ontoβ†’dom 𝐼)}
Distinct variable group:   𝑓,𝐺,𝑝
Allowed substitution hints:   𝐼(𝑓,𝑝)

Proof of Theorem eupths
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6897 . . . . 5 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = (iEdgβ€˜πΊ))
2 eupths.i . . . . 5 𝐼 = (iEdgβ€˜πΊ)
31, 2eqtr4di 2786 . . . 4 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = 𝐼)
43dmeqd 5908 . . 3 (𝑔 = 𝐺 β†’ dom (iEdgβ€˜π‘”) = dom 𝐼)
5 foeq3 6809 . . 3 (dom (iEdgβ€˜π‘”) = dom 𝐼 β†’ (𝑓:(0..^(β™―β€˜π‘“))–ontoβ†’dom (iEdgβ€˜π‘”) ↔ 𝑓:(0..^(β™―β€˜π‘“))–ontoβ†’dom 𝐼))
64, 5syl 17 . 2 (𝑔 = 𝐺 β†’ (𝑓:(0..^(β™―β€˜π‘“))–ontoβ†’dom (iEdgβ€˜π‘”) ↔ 𝑓:(0..^(β™―β€˜π‘“))–ontoβ†’dom 𝐼))
7 df-eupth 30021 . 2 EulerPaths = (𝑔 ∈ V ↦ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Trailsβ€˜π‘”)𝑝 ∧ 𝑓:(0..^(β™―β€˜π‘“))–ontoβ†’dom (iEdgβ€˜π‘”))})
86, 7fvmptopab 7474 1 (EulerPathsβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Trailsβ€˜πΊ)𝑝 ∧ 𝑓:(0..^(β™―β€˜π‘“))–ontoβ†’dom 𝐼)}
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534   class class class wbr 5148  {copab 5210  dom cdm 5678  β€“ontoβ†’wfo 6546  β€˜cfv 6548  (class class class)co 7420  0cc0 11139  ..^cfzo 13660  β™―chash 14322  iEdgciedg 28823  Trailsctrls 29517  EulerPathsceupth 30020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fo 6554  df-fv 6556  df-eupth 30021
This theorem is referenced by:  iseupth  30024
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