Theorem 2wlksot 26394

Description: The set of walks of length 2 (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 21-Feb-2018.)

Assertion

Ref	Expression
2wlksot	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉 2WalksOt 𝐸) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)})

Distinct variable groups:   𝐸,𝑎,𝑏,𝑡   𝑉,𝑎,𝑏,𝑡

Allowed substitution hints:   𝑋(𝑡,𝑎,𝑏)   𝑌(𝑡,𝑎,𝑏)

Proof of Theorem 2wlksot

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

Step	Hyp	Ref	Expression
1		elex 3185	. . 3 ⊢ (𝑉 ∈ 𝑋 → 𝑉 ∈ V)
2	1	adantr 480	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝑉 ∈ V)
3		elex 3185	. . 3 ⊢ (𝐸 ∈ 𝑌 → 𝐸 ∈ V)
4	3	adantl 481	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐸 ∈ V)
5		3xpexg 6859	. . . 4 ⊢ (𝑉 ∈ 𝑋 → ((𝑉 × 𝑉) × 𝑉) ∈ V)
6	5	adantr 480	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((𝑉 × 𝑉) × 𝑉) ∈ V)
7		rabexg 4739	. . 3 ⊢ (((𝑉 × 𝑉) × 𝑉) ∈ V → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)} ∈ V)
8	6, 7	syl 17	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)} ∈ V)
9		id 22	. . . . . . 7 ⊢ (𝑣 = 𝑉 → 𝑣 = 𝑉)
10	9, 9	xpeq12d 5064	. . . . . 6 ⊢ (𝑣 = 𝑉 → (𝑣 × 𝑣) = (𝑉 × 𝑉))
11	10, 9	xpeq12d 5064	. . . . 5 ⊢ (𝑣 = 𝑉 → ((𝑣 × 𝑣) × 𝑣) = ((𝑉 × 𝑉) × 𝑉))
12	11	adantr 480	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((𝑣 × 𝑣) × 𝑣) = ((𝑉 × 𝑉) × 𝑉))
13		simpl 472	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑣 = 𝑉)
14		oveq12 6558	. . . . . . . 8 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣 2WalksOnOt 𝑒) = (𝑉 2WalksOnOt 𝐸))
15	14	oveqd 6566	. . . . . . 7 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑎(𝑣 2WalksOnOt 𝑒)𝑏) = (𝑎(𝑉 2WalksOnOt 𝐸)𝑏))
16	15	eleq2d 2673	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑡 ∈ (𝑎(𝑣 2WalksOnOt 𝑒)𝑏) ↔ 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))
17	13, 16	rexeqbidv 3130	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (∃𝑏 ∈ 𝑣 𝑡 ∈ (𝑎(𝑣 2WalksOnOt 𝑒)𝑏) ↔ ∃𝑏 ∈ 𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))
18	13, 17	rexeqbidv 3130	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑡 ∈ (𝑎(𝑣 2WalksOnOt 𝑒)𝑏) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))
19	12, 18	rabeqbidv 3168	. . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑡 ∈ (𝑎(𝑣 2WalksOnOt 𝑒)𝑏)} = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)})
20		df-2wlksot 26386	. . 3 ⊢ 2WalksOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑡 ∈ (𝑎(𝑣 2WalksOnOt 𝑒)𝑏)})
21	19, 20	ovmpt2ga 6688	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)} ∈ V) → (𝑉 2WalksOt 𝐸) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)})
22	2, 4, 8, 21	syl3anc 1318	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉 2WalksOt 𝐸) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)})

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173   × cxp 5036  (class class class)co 6549   2WalksOt c2wlkot 26381   2WalksOnOt c2wlkonot 26382

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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-2wlksot 26386

This theorem is referenced by:  el2wlksoton  26405  el2wlksotot  26409