Theorem iseupth 16459

Description: The property "⟨𝐹, 𝑃⟩ is an Eulerian path on the graph 𝐺". An Eulerian path is defined as bijection 𝐹 from the edges to a set 0...(𝑁 − 1) and a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 0 ≤ 𝑘 < 𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘) to 𝑃(𝑘 + 1). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.)

Hypothesis

Ref	Expression
iseupth.i	⊢ 𝐼 = (iEdg‘𝐺)

Assertion

Ref	Expression
iseupth	⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))

Proof of Theorem iseupth

Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eupthv 16458	. 2 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
2		trlsv 16396	. . 3 ⊢ (𝐹(Trails‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
3	2	adantr 276	. 2 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
4		df-br 4112	. . . 4 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (EulerPaths‘𝐺))
5		iseupth.i	. . . . . . 7 ⊢ 𝐼 = (iEdg‘𝐺)
6	5	eupthsg 16457	. . . . . 6 ⊢ (𝐺 ∈ V → (EulerPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)})
7	6	3ad2ant1 1045	. . . . 5 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (EulerPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)})
8	7	eleq2d 2304	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (⟨𝐹, 𝑃⟩ ∈ (EulerPaths‘𝐺) ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)}))
9	4, 8	bitrid 192	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(EulerPaths‘𝐺)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)}))
10		breq1 4114	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓(Trails‘𝐺)𝑝 ↔ 𝐹(Trails‘𝐺)𝑝))
11		id 19	. . . . . . 7 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
12		fveq2 5672	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
13	12	oveq2d 6068	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹)))
14		eqidd 2235	. . . . . . 7 ⊢ (𝑓 = 𝐹 → dom 𝐼 = dom 𝐼)
15	11, 13, 14	foeq123d 5609	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼 ↔ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))
16	10, 15	anbi12d 473	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼) ↔ (𝐹(Trails‘𝐺)𝑝 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼)))
17		breq2 4115	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝐹(Trails‘𝐺)𝑝 ↔ 𝐹(Trails‘𝐺)𝑃))
18	17	anbi1d 465	. . . . 5 ⊢ (𝑝 = 𝑃 → ((𝐹(Trails‘𝐺)𝑝 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼)))
19	16, 18	opelopabg 4388	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑃 ∈ V) → (⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)} ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼)))
20	19	3adant1 1042	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)} ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼)))
21	9, 20	bitrd 188	. 2 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼)))
22	1, 3, 21	pm5.21nii 712	1 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  Vcvv 2815  ⟨cop 3694   class class class wbr 4111  {copab 4172  dom cdm 4751  –onto→wfo 5352  ‘cfv 5354  (class class class)co 6052  0cc0 8129  ..^cfzo 10480  ♯chash 11142  iEdgciedg 16025  Trailsctrls 16392  EulerPathsceupth 16454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-ifp 987  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-1o 6649  df-er 6769  df-map 6886  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-z 9580  df-dec 9713  df-uz 9857  df-fz 10346  df-fzo 10481  df-ihash 11143  df-word 11229  df-ndx 13232  df-slot 13233  df-base 13235  df-edgf 16017  df-vtx 16026  df-iedg 16027  df-wlks 16330  df-trls 16393  df-eupth 16455

This theorem is referenced by:  iseupthf1o  16460  eupthfi  16463  eupthistrl  16466