Theorem eupths 30275

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 6834	. . . . 5 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2		eupths.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	eqtr4di 2789	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
4	3	dmeqd 5854	. . 3 ⊢ (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom 𝐼)
5		foeq3 6744	. . 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 30273	. 2 ⊢ EulerPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})
8	6, 7	fvmptopab 7413	1 ⊢ (EulerPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   class class class wbr 5098  {copab 5160  dom cdm 5624  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7358  0cc0 11026  ..^cfzo 13570  ♯chash 14253  iEdgciedg 29070  Trailsctrls 29762  EulerPathsceupth 30272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fo 6498  df-fv 6500  df-eupth 30273

This theorem is referenced by:  iseupth  30276