Theorem wlkson 29909

Description: The set of walks between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 30-Dec-2020.) (Revised by AV, 22-Mar-2021.)

Hypothesis

Ref	Expression
wlkson.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
wlkson	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵)})

Distinct variable groups:   𝐴,𝑓,𝑝   𝐵,𝑓,𝑝   𝑓,𝐺,𝑝   𝑓,𝑉,𝑝

Proof of Theorem wlkson

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

Step	Hyp	Ref	Expression
1		wlkson.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	1vgrex 29257	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ V)
3	2	adantr 485	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V)
4		simpl 487	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
5	4, 1	eleqtrdi 2875	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ (Vtx‘𝐺))
6		simpr 489	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
7	6, 1	eleqtrdi 2875	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ (Vtx‘𝐺))
8		eqeq2 2777	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑝‘0) = 𝑎 ↔ (𝑝‘0) = 𝐴))
9		eqeq2 2777	. . . 4 ⊢ (𝑏 = 𝐵 → ((𝑝‘(♯‘𝑓)) = 𝑏 ↔ (𝑝‘(♯‘𝑓)) = 𝐵))
10	8, 9	bi2anan9 649	. . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ↔ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵)))
11		biidd 265	. . 3 ⊢ (𝑔 = 𝐺 → (((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ↔ ((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)))
12		df-wlkson 29855	. . . 4 ⊢ WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)}))
13		eqid 2765	. . . . . 6 ⊢ (Vtx‘𝑔) = (Vtx‘𝑔)
14		3anass 1109	. . . . . . . 8 ⊢ ((𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ↔ (𝑓(Walks‘𝑔)𝑝 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)))
15	14	biancomi 467	. . . . . . 7 ⊢ ((𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ↔ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝))
16	15	opabbii 5171	. . . . . 6 ⊢ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)} = {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝)}
17	13, 13, 16	mpoeq123i 7476	. . . . 5 ⊢ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)}) = (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝)})
18	17	mpteq2i 5200	. . . 4 ⊢ (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)})) = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝)}))
19	12, 18	eqtri 2788	. . 3 ⊢ WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝)}))
20	3, 5, 7, 10, 11, 19	mptmpoopabbrd 8066	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝)})
21		ancom 465	. . . 4 ⊢ ((((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵)))
22		3anass 1109	. . . 4 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵)))
23	21, 22	bitr4i 281	. . 3 ⊢ ((((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵))
24	23	opabbii 5171	. 2 ⊢ {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵)}
25	20, 24	eqtrdi 2816	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵)})

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  Vcvv 3457   class class class wbr 5104  {copab 5166   ↦ cmpt 5185  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  0cc0 11088  ♯chash 14354  Vtxcvtx 29251  Walkscwlks 29851  WalksOncwlkson 29852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  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-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-wlkson 29855

This theorem is referenced by:  iswlkon  29910  wlkonprop  29911