Theorem upgrspthswlk 27797

Description: The set of simple paths in a pseudograph, expressed as walk. Notice that this theorem would not hold for arbitrary hypergraphs, since a walk with distinct vertices does not need to be a trail: let E = { p₀, p₁, p₂ } be a hyperedge, then ( p₀, e, p₁, e, p₂ ) is walk with distinct vertices, but not with distinct edges. Therefore, E is not a trail and, by definition, also no path. (Contributed by AV, 11-Jan-2021.) (Proof shortened by AV, 17-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)

Assertion

Ref	Expression
upgrspthswlk	⊢ (𝐺 ∈ UPGraph → (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)})

Distinct variable group:   𝑓,𝐺,𝑝

Proof of Theorem upgrspthswlk

Step	Hyp	Ref	Expression
1		spthsfval 27781	. 2 ⊢ (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)}
2		istrl 27756	. . . . . 6 ⊢ (𝑓(Trails‘𝐺)𝑝 ↔ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑓))
3		upgrwlkdvde 27796	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝) → Fun ◡𝑓)
4	3	3exp 1121	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → (𝑓(Walks‘𝐺)𝑝 → (Fun ◡𝑝 → Fun ◡𝑓)))
5	4	com23 86	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (Fun ◡𝑝 → (𝑓(Walks‘𝐺)𝑝 → Fun ◡𝑓)))
6	5	imp 410	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ Fun ◡𝑝) → (𝑓(Walks‘𝐺)𝑝 → Fun ◡𝑓))
7	6	pm4.71d 565	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ Fun ◡𝑝) → (𝑓(Walks‘𝐺)𝑝 ↔ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑓)))
8	2, 7	bitr4id 293	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ Fun ◡𝑝) → (𝑓(Trails‘𝐺)𝑝 ↔ 𝑓(Walks‘𝐺)𝑝))
9	8	ex 416	. . . 4 ⊢ (𝐺 ∈ UPGraph → (Fun ◡𝑝 → (𝑓(Trails‘𝐺)𝑝 ↔ 𝑓(Walks‘𝐺)𝑝)))
10	9	pm5.32rd 581	. . 3 ⊢ (𝐺 ∈ UPGraph → ((𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)))
11	10	opabbidv 5109	. 2 ⊢ (𝐺 ∈ UPGraph → {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)})
12	1, 11	syl5eq 2786	1 ⊢ (𝐺 ∈ UPGraph → (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110   class class class wbr 5043  {copab 5105  ^◡ccnv 5539  Fun wfun 6363  ‘cfv 6369  UPGraphcupgr 27143  Walkscwlks 27656  Trailsctrls 27750  SPathscspths 27772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ifp 1064  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-int 4850  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-1st 7750  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-1o 8191  df-2o 8192  df-oadd 8195  df-er 8380  df-map 8499  df-pm 8500  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-dju 9500  df-card 9538  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-nn 11814  df-2 11876  df-n0 12074  df-xnn0 12146  df-z 12160  df-uz 12422  df-fz 13079  df-fzo 13222  df-hash 13880  df-word 14053  df-edg 27111  df-uhgr 27121  df-upgr 27145  df-wlks 27659  df-trls 27752  df-spths 27776

This theorem is referenced by:  upgrwlkdvspth  27798