Theorem upgrwlkdvspth 26635

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 26634. (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.

Step	Hyp	Ref	Expression
1		3simpc 1060	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝑃) → (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝑃))
2		upgrspthswlk 26634	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)})
3	2	3ad2ant1 1082	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝑃) → (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)})
4	3	breqd 4664	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝑃) → (𝐹(SPaths‘𝐺)𝑃 ↔ 𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)}𝑃))
5		wlkv 26508	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
6		3simpc 1060	. . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹 ∈ V ∧ 𝑃 ∈ V))
7	5, 6	syl 17	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ V ∧ 𝑃 ∈ V))
8	7	3ad2ant2 1083	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝑃) → (𝐹 ∈ V ∧ 𝑃 ∈ V))
9		breq12 4658	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓(Walks‘𝐺)𝑝 ↔ 𝐹(Walks‘𝐺)𝑃))
10		cnveq 5296	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ◡𝑝 = ◡𝑃)
11	10	funeqd 5910	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (Fun ◡𝑝 ↔ Fun ◡𝑃))
12	11	adantl 482	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (Fun ◡𝑝 ↔ Fun ◡𝑃))
13	9, 12	anbi12d 747	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝) ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝑃)))
14		eqid 2622	. . . . 5 ⊢ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)}
15	13, 14	brabga 4989	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)}𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝑃)))
16	8, 15	syl 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)))
17	4, 16	bitrd 268	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝑃) → (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝑃)))
18	1, 17	mpbird 247	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝑃) → 𝐹(SPaths‘𝐺)𝑃)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200   class class class wbr 4653  {copab 4712  ^◡ccnv 5113  Fun wfun 5882  ‘cfv 5888   UPGraph cupgr 25975  Walkscwlks 26492  SPathscspths 26609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-wlks 26495  df-trls 26589  df-spths 26613

This theorem is referenced by: (None)