Theorem spthon 26112

Description: The set of simple paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)

Assertion

Ref	Expression
spthon	⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑉 SPathOn 𝐸)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)})

Distinct variable groups:   𝑓,𝐸,𝑝   𝑓,𝑉,𝑝   𝑓,𝑋,𝑝   𝑓,𝑌,𝑝   𝐴,𝑓,𝑝   𝐵,𝑓,𝑝

Proof of Theorem spthon

Dummy variables 𝑎 𝑏 𝑒 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3185	. . . . 5 ⊢ (𝑉 ∈ 𝑋 → 𝑉 ∈ V)
2	1	ad2antrr 758	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝑉 ∈ V)
3		elex 3185	. . . . . 6 ⊢ (𝐸 ∈ 𝑌 → 𝐸 ∈ V)
4	3	adantl 481	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐸 ∈ V)
5	4	adantr 480	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐸 ∈ V)
6		id 22	. . . . . . 7 ⊢ (𝑉 ∈ 𝑋 → 𝑉 ∈ 𝑋)
7	6	ancli 572	. . . . . 6 ⊢ (𝑉 ∈ 𝑋 → (𝑉 ∈ 𝑋 ∧ 𝑉 ∈ 𝑋))
8	7	ad2antrr 758	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝑉 ∈ 𝑋 ∧ 𝑉 ∈ 𝑋))
9		mpt2exga 7135	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑉 ∈ 𝑋) → (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)}) ∈ V)
10	8, 9	syl 17	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)}) ∈ V)
11		simpl 472	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑣 = 𝑉)
12		oveq12 6558	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣 WalkOn 𝑒) = (𝑉 WalkOn 𝐸))
13	12	oveqd 6566	. . . . . . . . 9 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑎(𝑣 WalkOn 𝑒)𝑏) = (𝑎(𝑉 WalkOn 𝐸)𝑏))
14	13	breqd 4594	. . . . . . . 8 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ↔ 𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝))
15		oveq12 6558	. . . . . . . . 9 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣 SPaths 𝑒) = (𝑉 SPaths 𝐸))
16	15	breqd 4594	. . . . . . . 8 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑓(𝑣 SPaths 𝑒)𝑝 ↔ 𝑓(𝑉 SPaths 𝐸)𝑝))
17	14, 16	anbi12d 743	. . . . . . 7 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ∧ 𝑓(𝑣 SPaths 𝑒)𝑝) ↔ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)))
18	17	opabbidv 4648	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ∧ 𝑓(𝑣 SPaths 𝑒)𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)})
19	11, 11, 18	mpt2eq123dv 6615	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ∧ 𝑓(𝑣 SPaths 𝑒)𝑝)}) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)}))
20		df-spthon 26045	. . . . 5 ⊢ SPathOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ∧ 𝑓(𝑣 SPaths 𝑒)𝑝)}))
21	19, 20	ovmpt2ga 6688	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)}) ∈ V) → (𝑉 SPathOn 𝐸) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)}))
22	2, 5, 10, 21	syl3anc 1318	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝑉 SPathOn 𝐸) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)}))
23	22	oveqd 6566	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑉 SPathOn 𝐸)𝐵) = (𝐴(𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)})𝐵))
24		simpl 472	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
25	24	adantl 481	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐴 ∈ 𝑉)
26		simprr 792	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
27		ancom 465	. . . . . . 7 ⊢ ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝) ↔ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ 𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝))
28	27	a1i 11	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝) ↔ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ 𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝)))
29	28	opabbidv 4648	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ 𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝)})
30		spthispth 26103	. . . . . . . 8 ⊢ (𝑓(𝑉 SPaths 𝐸)𝑝 → 𝑓(𝑉 Paths 𝐸)𝑝)
31		pthistrl 26102	. . . . . . . 8 ⊢ (𝑓(𝑉 Paths 𝐸)𝑝 → 𝑓(𝑉 Trails 𝐸)𝑝)
32		trliswlk 26069	. . . . . . . 8 ⊢ (𝑓(𝑉 Trails 𝐸)𝑝 → 𝑓(𝑉 Walks 𝐸)𝑝)
33	30, 31, 32	3syl 18	. . . . . . 7 ⊢ (𝑓(𝑉 SPaths 𝐸)𝑝 → 𝑓(𝑉 Walks 𝐸)𝑝)
34	33	wlkres 26050	. . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ 𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝)} ∈ V)
35	1, 3, 34	syl2an 493	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ 𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝)} ∈ V)
36	29, 35	eqeltrd 2688	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)} ∈ V)
37	36	adantr 480	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)} ∈ V)
38		oveq12 6558	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎(𝑉 WalkOn 𝐸)𝑏) = (𝐴(𝑉 WalkOn 𝐸)𝐵))
39	38	breqd 4594	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ↔ 𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝))
40	39	anbi1d 737	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝) ↔ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)))
41	40	opabbidv 4648	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)})
42		eqid 2610	. . . 4 ⊢ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)}) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)})
43	41, 42	ovmpt2ga 6688	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)} ∈ V) → (𝐴(𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)})𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)})
44	25, 26, 37, 43	syl3anc 1318	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)})𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)})
45	23, 44	eqtrd 2644	1 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑉 SPathOn 𝐸)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ 𝑓(𝑉 SPaths 𝐸)𝑝)})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  {copab 4642  (class class class)co 6549   ↦ cmpt2 6551   Walks cwalk 26026   Trails ctrail 26027   Paths cpath 26028   SPaths cspath 26029   WalkOn cwlkon 26030   SPathOn cspthon 26033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-wlk 26036  df-trail 26037  df-pth 26038  df-spth 26039  df-spthon 26045

This theorem is referenced by:  isspthon  26113  spthonprp  26115