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Theorem eupths 29147
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 6843 . . . . 5 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = (iEdgβ€˜πΊ))
2 eupths.i . . . . 5 𝐼 = (iEdgβ€˜πΊ)
31, 2eqtr4di 2795 . . . 4 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = 𝐼)
43dmeqd 5862 . . 3 (𝑔 = 𝐺 β†’ dom (iEdgβ€˜π‘”) = dom 𝐼)
5 foeq3 6755 . . 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 29145 . 2 EulerPaths = (𝑔 ∈ V ↦ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Trailsβ€˜π‘”)𝑝 ∧ 𝑓:(0..^(β™―β€˜π‘“))–ontoβ†’dom (iEdgβ€˜π‘”))})
86, 7fvmptopab 7412 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 397   = wceq 1542   class class class wbr 5106  {copab 5168  dom cdm 5634  β€“ontoβ†’wfo 6495  β€˜cfv 6497  (class class class)co 7358  0cc0 11052  ..^cfzo 13568  β™―chash 14231  iEdgciedg 27951  Trailsctrls 28641  EulerPathsceupth 29144
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 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fo 6503  df-fv 6505  df-eupth 29145
This theorem is referenced by:  iseupth  29148
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