Theorem wlkson 40864

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(1Walks‘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 25679	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ V)
3	2	adantr 480	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V)
4		simpl 472	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
5	4, 1	syl6eleq 2698	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ (Vtx‘𝐺))
6		simpr 476	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
7	6, 1	syl6eleq 2698	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ (Vtx‘𝐺))
8		1wlksv 40824	. . . 4 ⊢ {⟨𝑓, 𝑝⟩ ∣ 𝑓(1Walks‘𝐺)𝑝} ∈ V
9	8	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → {⟨𝑓, 𝑝⟩ ∣ 𝑓(1Walks‘𝐺)𝑝} ∈ V)
10		simpr 476	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑓(1Walks‘𝐺)𝑝) → 𝑓(1Walks‘𝐺)𝑝)
11		eqeq2 2621	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑝‘0) = 𝑎 ↔ (𝑝‘0) = 𝐴))
12		eqeq2 2621	. . . 4 ⊢ (𝑏 = 𝐵 → ((𝑝‘(#‘𝑓)) = 𝑏 ↔ (𝑝‘(#‘𝑓)) = 𝐵))
13	11, 12	bi2anan9 913	. . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ↔ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)))
14		biidd 251	. . 3 ⊢ (𝑔 = 𝐺 → (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ↔ ((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)))
15		df-wlkson 40802	. . . 4 ⊢ WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)}))
16		eqid 2610	. . . . . 6 ⊢ (Vtx‘𝑔) = (Vtx‘𝑔)
17		3anass 1035	. . . . . . . 8 ⊢ ((𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ↔ (𝑓(1Walks‘𝑔)𝑝 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)))
18		ancom 465	. . . . . . . 8 ⊢ ((𝑓(1Walks‘𝑔)𝑝 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)) ↔ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(1Walks‘𝑔)𝑝))
19	17, 18	bitri 263	. . . . . . 7 ⊢ ((𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ↔ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(1Walks‘𝑔)𝑝))
20	19	opabbii 4649	. . . . . 6 ⊢ {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)} = {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(1Walks‘𝑔)𝑝)}
21	16, 16, 20	mpt2eq123i 6616	. . . . 5 ⊢ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)}) = (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(1Walks‘𝑔)𝑝)})
22	21	mpteq2i 4669	. . . 4 ⊢ (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘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(1Walks‘𝑔)𝑝)}))
23	15, 22	eqtri 2632	. . 3 ⊢ WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(1Walks‘𝑔)𝑝)}))
24	3, 5, 7, 9, 10, 13, 14, 23	mptmpt2opabbrd 40335	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵) ∧ 𝑓(1Walks‘𝐺)𝑝)})
25		ancom 465	. . . 4 ⊢ ((((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵) ∧ 𝑓(1Walks‘𝐺)𝑝) ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)))
26		3anass 1035	. . . 4 ⊢ ((𝑓(1Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵) ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)))
27	25, 26	bitr4i 266	. . 3 ⊢ ((((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵) ∧ 𝑓(1Walks‘𝐺)𝑝) ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵))
28	27	opabbii 4649	. 2 ⊢ {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵) ∧ 𝑓(1Walks‘𝐺)𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)}
29	24, 28	syl6eq 2660	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)})

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  {copab 4642   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  #chash 12979  Vtxcvtx 25673  1Walksc1wlks 40796  WalksOncwlkson 40798

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-ifp 1007  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-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-1wlks 40800  df-wlkson 40802

This theorem is referenced by:  iswlkOn  40865  wlkOnprop  40866