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Theorem upgriseupth 29728
Description: The property "⟨𝐹, π‘ƒβŸ© is an Eulerian path on the pseudograph 𝐺". (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.)
Hypotheses
Ref Expression
eupths.i 𝐼 = (iEdgβ€˜πΊ)
upgriseupth.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
upgriseupth (𝐺 ∈ UPGraph β†’ (𝐹(EulerPathsβ€˜πΊ)𝑃 ↔ (𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})))
Distinct variable groups:   π‘˜,𝐹   π‘˜,𝐺   π‘˜,𝐼   𝑃,π‘˜   π‘˜,𝑉
Proof of Theorem upgriseupth
StepHypRef Expression
1 eupths.i . . . 4 𝐼 = (iEdgβ€˜πΊ)
21iseupthf1o 29723 . . 3 (𝐹(EulerPathsβ€˜πΊ)𝑃 ↔ (𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼))
32a1i 11 . 2 (𝐺 ∈ UPGraph β†’ (𝐹(EulerPathsβ€˜πΊ)𝑃 ↔ (𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼)))
4 upgriseupth.v . . . 4 𝑉 = (Vtxβ€˜πΊ)
54, 1upgriswlk 29166 . . 3 (𝐺 ∈ UPGraph β†’ (𝐹(Walksβ€˜πΊ)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})))
65anbi1d 629 . 2 (𝐺 ∈ UPGraph β†’ ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼) ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼)))
7 simpr 484 . . . . 5 (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼) β†’ 𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼)
8 simpl2 1191 . . . . 5 (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼) β†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
9 simpl3 1192 . . . . 5 (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼) β†’ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})
107, 8, 93jca 1127 . . . 4 (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼) β†’ (𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}))
11 f1of 6833 . . . . . . 7 (𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼 β†’ 𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐼)
12 iswrdi 14473 . . . . . . 7 (𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐼 β†’ 𝐹 ∈ Word dom 𝐼)
1311, 12syl 17 . . . . . 6 (𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼 β†’ 𝐹 ∈ Word dom 𝐼)
14133anim1i 1151 . . . . 5 ((𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) β†’ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}))
15 simp1 1135 . . . . 5 ((𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) β†’ 𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼)
1614, 15jca 511 . . . 4 ((𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) β†’ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼))
1710, 16impbii 208 . . 3 (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼) ↔ (𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}))
1817a1i 11 . 2 (𝐺 ∈ UPGraph β†’ (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼) ↔ (𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})))
193, 6, 183bitrd 305 1 (𝐺 ∈ UPGraph β†’ (𝐹(EulerPathsβ€˜πΊ)𝑃 ↔ (𝐹:(0..^(β™―β€˜πΉ))–1-1-ontoβ†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  {cpr 4630   class class class wbr 5148  dom cdm 5676  βŸΆwf 6539  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7412  0cc0 11114  1c1 11115   + caddc 11117  ...cfz 13489  ..^cfzo 13632  β™―chash 14295  Word cword 14469  Vtxcvtx 28524  iEdgciedg 28525  UPGraphcupgr 28608  Walkscwlks 29121  EulerPathsceupth 29718
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1061  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-2o 8471  df-oadd 8474  df-er 8707  df-map 8826  df-pm 8827  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-dju 9900  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-n0 12478  df-xnn0 12550  df-z 12564  df-uz 12828  df-fz 13490  df-fzo 13633  df-hash 14296  df-word 14470  df-edg 28576  df-uhgr 28586  df-upgr 28610  df-wlks 29124  df-trls 29217  df-eupth 29719
This theorem is referenced by:  upgreupthi  29729
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