Theorem wlkson 29748

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 29096	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ V)
3	2	adantr 481	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V)
4		simpl 483	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
5	4, 1	eleqtrdi 2850	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ (Vtx‘𝐺))
6		simpr 485	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
7	6, 1	eleqtrdi 2850	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ (Vtx‘𝐺))
8		eqeq2 2752	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑝‘0) = 𝑎 ↔ (𝑝‘0) = 𝐴))
9		eqeq2 2752	. . . 4 ⊢ (𝑏 = 𝐵 → ((𝑝‘(♯‘𝑓)) = 𝑏 ↔ (𝑝‘(♯‘𝑓)) = 𝐵))
10	8, 9	bi2anan9 644	. . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ↔ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵)))
11		biidd 263	. . 3 ⊢ (𝑔 = 𝐺 → (((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ↔ ((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)))
12		df-wlkson 29694	. . . 4 ⊢ WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)}))
13		eqid 2740	. . . . . 6 ⊢ (Vtx‘𝑔) = (Vtx‘𝑔)
14		3anass 1100	. . . . . . . 8 ⊢ ((𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ↔ (𝑓(Walks‘𝑔)𝑝 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)))
15	14	biancomi 463	. . . . . . 7 ⊢ ((𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ↔ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝))
16	15	opabbii 5146	. . . . . 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 7439	. . . . 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 5175	. . . 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 2763	. . 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 8029	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝)})
21		ancom 461	. . . 4 ⊢ ((((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵)))
22		3anass 1100	. . . 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 279	. . 3 ⊢ ((((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵))
24	23	opabbii 5146	. 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 2791	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 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  Vcvv 3432   class class class wbr 5079  {copab 5141   ↦ cmpt 5160  ‘cfv 6492  (class class class)co 7363   ∈ cmpo 7365  0cc0 11036  ♯chash 14290  Vtxcvtx 29090  Walkscwlks 29690  WalksOncwlkson 29691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-wlkson 29694

This theorem is referenced by:  iswlkon  29749  wlkonprop  29750