Theorem eupths 30232

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 = (iEdg‘𝐺)

Assertion

Ref	Expression
eupths	⊢ (EulerPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→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.

Step	Hyp	Ref	Expression
1		fveq2 6920	. . . . 5 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2		eupths.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	eqtr4di 2798	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
4	3	dmeqd 5930	. . 3 ⊢ (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom 𝐼)
5		foeq3 6832	. . 3 ⊢ (dom (iEdg‘𝑔) = dom 𝐼 → (𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔) ↔ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼))
6	4, 5	syl 17	. 2 ⊢ (𝑔 = 𝐺 → (𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔) ↔ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼))
7		df-eupth 30230	. 2 ⊢ EulerPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})
8	6, 7	fvmptopab 7504	1 ⊢ (EulerPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   class class class wbr 5166  {copab 5228  dom cdm 5700  –onto→wfo 6571  ‘cfv 6573  (class class class)co 7448  0cc0 11184  ..^cfzo 13711  ♯chash 14379  iEdgciedg 29032  Trailsctrls 29726  EulerPathsceupth 30229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fo 6579  df-fv 6581  df-eupth 30230

This theorem is referenced by:  iseupth  30233