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Theorem eupths 29948
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 6882 . . . . 5 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = (iEdgβ€˜πΊ))
2 eupths.i . . . . 5 𝐼 = (iEdgβ€˜πΊ)
31, 2eqtr4di 2782 . . . 4 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = 𝐼)
43dmeqd 5896 . . 3 (𝑔 = 𝐺 β†’ dom (iEdgβ€˜π‘”) = dom 𝐼)
5 foeq3 6794 . . 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 29946 . 2 EulerPaths = (𝑔 ∈ V ↦ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Trailsβ€˜π‘”)𝑝 ∧ 𝑓:(0..^(β™―β€˜π‘“))–ontoβ†’dom (iEdgβ€˜π‘”))})
86, 7fvmptopab 7456 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 1533   class class class wbr 5139  {copab 5201  dom cdm 5667  β€“ontoβ†’wfo 6532  β€˜cfv 6534  (class class class)co 7402  0cc0 11107  ..^cfzo 13628  β™―chash 14291  iEdgciedg 28751  Trailsctrls 29442  EulerPathsceupth 29945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fo 6540  df-fv 6542  df-eupth 29946
This theorem is referenced by:  iseupth  29949
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