Theorem upgrspthswlk 29591

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 29575	. 2 ⊢ (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)}
2		istrl 29549	. . . . . 6 ⊢ (𝑓(Trails‘𝐺)𝑝 ↔ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑓))
3		upgrwlkdvde 29590	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝) → Fun ◡𝑓)
4	3	3exp 1116	. . . . . . . . 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 405	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ Fun ◡𝑝) → (𝑓(Walks‘𝐺)𝑝 → Fun ◡𝑓))
7	6	pm4.71d 560	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ Fun ◡𝑝) → (𝑓(Walks‘𝐺)𝑝 ↔ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑓)))
8	2, 7	bitr4id 289	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ Fun ◡𝑝) → (𝑓(Trails‘𝐺)𝑝 ↔ 𝑓(Walks‘𝐺)𝑝))
9	8	ex 411	. . . 4 ⊢ (𝐺 ∈ UPGraph → (Fun ◡𝑝 → (𝑓(Trails‘𝐺)𝑝 ↔ 𝑓(Walks‘𝐺)𝑝)))
10	9	pm5.32rd 576	. . 3 ⊢ (𝐺 ∈ UPGraph → ((𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)))
11	10	opabbidv 5210	. 2 ⊢ (𝐺 ∈ UPGraph → {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)})
12	1, 11	eqtrid 2777	1 ⊢ (𝐺 ∈ UPGraph → (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   class class class wbr 5144  {copab 5206  ^◡ccnv 5672  Fun wfun 6537  ‘cfv 6543  UPGraphcupgr 28932  Walkscwlks 29449  Trailsctrls 29543  SPathscspths 29566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  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 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-map 8840  df-pm 8841  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-dju 9919  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-n0 12498  df-xnn0 12570  df-z 12584  df-uz 12848  df-fz 13512  df-fzo 13655  df-hash 14317  df-word 14492  df-edg 28900  df-uhgr 28910  df-upgr 28934  df-wlks 29452  df-trls 29545  df-spths 29570

This theorem is referenced by:  upgrwlkdvspth  29592