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Theorem upgrwlkdvspth 26866
Description: A walk consisting of different vertices is a simple path. Notice that this theorem would not hold for arbitrary hypergraphs, see the counterexample given in the comment of upgrspthswlk 26865. (Contributed by Alexander van der Vekens, 27-Oct-2017.) (Revised by AV, 17-Jan-2021.)
Assertion
Ref Expression
upgrwlkdvspth ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → 𝐹(SPaths‘𝐺)𝑃)

Proof of Theorem upgrwlkdvspth
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpc 1147 . 2 ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃))
2 upgrspthswlk 26865 . . . . 5 (𝐺 ∈ UPGraph → (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)})
323ad2ant1 1128 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)})
43breqd 4815 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → (𝐹(SPaths‘𝐺)𝑃𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)}𝑃))
5 wlkv 26739 . . . . . 6 (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
6 3simpc 1147 . . . . . 6 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹 ∈ V ∧ 𝑃 ∈ V))
75, 6syl 17 . . . . 5 (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ V ∧ 𝑃 ∈ V))
873ad2ant2 1129 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → (𝐹 ∈ V ∧ 𝑃 ∈ V))
9 breq12 4809 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓(Walks‘𝐺)𝑝𝐹(Walks‘𝐺)𝑃))
10 cnveq 5451 . . . . . . . 8 (𝑝 = 𝑃𝑝 = 𝑃)
1110funeqd 6071 . . . . . . 7 (𝑝 = 𝑃 → (Fun 𝑝 ↔ Fun 𝑃))
1211adantl 473 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (Fun 𝑝 ↔ Fun 𝑃))
139, 12anbi12d 749 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝) ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃)))
14 eqid 2760 . . . . 5 {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)}
1513, 14brabga 5139 . . . 4 ((𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)}𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃)))
168, 15syl 17 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → (𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)}𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃)))
174, 16bitrd 268 . 2 ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃)))
181, 17mpbird 247 1 ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → 𝐹(SPaths‘𝐺)𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  Vcvv 3340   class class class wbr 4804  {copab 4864  ccnv 5265  Fun wfun 6043  cfv 6049  UPGraphcupgr 26195  Walkscwlks 26723  SPathscspths 26840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1051  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-er 7913  df-map 8027  df-pm 8028  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-n0 11505  df-xnn0 11576  df-z 11590  df-uz 11900  df-fz 12540  df-fzo 12680  df-hash 13332  df-word 13505  df-edg 26160  df-uhgr 26173  df-upgr 26197  df-wlks 26726  df-trls 26820  df-spths 26844
This theorem is referenced by: (None)
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