Theorem upgrspthswlk 30324

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 30305	. 2 ⊢ (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)}
2		istrl 30279	. . . . . 6 ⊢ (𝑓(Trails‘𝐺)𝑝 ↔ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑓))
3		upgrwlkdvde 30323	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝) → Fun ◡𝑓)
4	3	3exp 1137	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → (𝑓(Walks‘𝐺)𝑝 → (Fun ◡𝑝 → Fun ◡𝑓)))
5	4	com23 87	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (Fun ◡𝑝 → (𝑓(Walks‘𝐺)𝑝 → Fun ◡𝑓)))
6	5	imp 412	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ Fun ◡𝑝) → (𝑓(Walks‘𝐺)𝑝 → Fun ◡𝑓))
7	6	pm4.71d 571	. . . . . 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 418	. . . 4 ⊢ (𝐺 ∈ UPGraph → (Fun ◡𝑝 → (𝑓(Trails‘𝐺)𝑝 ↔ 𝑓(Walks‘𝐺)𝑝)))
10	9	pm5.32rd 589	. . 3 ⊢ (𝐺 ∈ UPGraph → ((𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)))
11	10	opabbidv 5171	. 2 ⊢ (𝐺 ∈ UPGraph → {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)})
12	1, 11	eqtrid 2808	1 ⊢ (𝐺 ∈ UPGraph → (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   class class class wbr 5103  {copab 5167  ^◡ccnv 5650  Fun wfun 6532  ‘cfv 6538  UPGraphcupgr 29658  Walkscwlks 30177  Trailsctrls 30273  SPathscspths 30296

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-er 8717  df-map 8849  df-pm 8850  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-dju 9982  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-n0 12607  df-xnn0 12680  df-z 12694  df-uz 12966  df-fz 13640  df-fzo 13789  df-hash 14475  df-word 14659  df-edg 29626  df-uhgr 29636  df-upgr 29660  df-wlks 30180  df-trls 30275  df-spths 30300

This theorem is used by:  upgrwlkdvspth  30325