Theorem 1pthon2v 30182

Description: For each pair of adjacent vertices there is a path of length 1 from one vertex to the other in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)

Hypotheses

Ref	Expression
1pthon2v.v	⊢ 𝑉 = (Vtx‘𝐺)
1pthon2v.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
1pthon2v	⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)

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

Allowed substitution hints:   𝐸(𝑒,𝑓,𝑝)   𝑉(𝑓,𝑝)

Proof of Theorem 1pthon2v

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
2	1	anim2i 617	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐺 ∈ UHGraph ∧ 𝐴 ∈ 𝑉))
3	2	3adant3 1131	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒) → (𝐺 ∈ UHGraph ∧ 𝐴 ∈ 𝑉))
4	3	adantl 481	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → (𝐺 ∈ UHGraph ∧ 𝐴 ∈ 𝑉))
5		1pthon2v.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
6	5	0pthonv 30158	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝)
7	4, 6	simpl2im 503	. . . 4 ⊢ ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝)
8		oveq2 7439	. . . . . . . 8 ⊢ (𝐵 = 𝐴 → (𝐴(PathsOn‘𝐺)𝐵) = (𝐴(PathsOn‘𝐺)𝐴))
9	8	eqcoms 2743	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴(PathsOn‘𝐺)𝐵) = (𝐴(PathsOn‘𝐺)𝐴))
10	9	breqd 5159	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝))
11	10	2exbidv 1922	. . . . 5 ⊢ (𝐴 = 𝐵 → (∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝))
12	11	adantr 480	. . . 4 ⊢ ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → (∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝))
13	7, 12	mpbird 257	. . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)
14	13	ex 412	. 2 ⊢ (𝐴 = 𝐵 → ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
15		1pthon2v.e	. . . . . . . . . . 11 ⊢ 𝐸 = (Edg‘𝐺)
16	15	eleq2i 2831	. . . . . . . . . 10 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
17		eqid 2735	. . . . . . . . . . 11 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
18	17	uhgredgiedgb 29158	. . . . . . . . . 10 ⊢ (𝐺 ∈ UHGraph → (𝑒 ∈ (Edg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖)))
19	16, 18	bitrid 283	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → (𝑒 ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖)))
20	19	3ad2ant1 1132	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (𝑒 ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖)))
21		s1cli 14640	. . . . . . . . . . . 12 ⊢ ⟨“𝑖”⟩ ∈ Word V
22		s2cli 14916	. . . . . . . . . . . 12 ⊢ ⟨“𝐴𝐵”⟩ ∈ Word V
23	21, 22	pm3.2i 470	. . . . . . . . . . 11 ⊢ (⟨“𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵”⟩ ∈ Word V)
24		eqid 2735	. . . . . . . . . . . 12 ⊢ ⟨“𝐴𝐵”⟩ = ⟨“𝐴𝐵”⟩
25		eqid 2735	. . . . . . . . . . . 12 ⊢ ⟨“𝑖”⟩ = ⟨“𝑖”⟩
26		simpl2l 1225	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → 𝐴 ∈ 𝑉)
27		simpl2r 1226	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → 𝐵 ∈ 𝑉)
28		eqneqall 2949	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
29	28	com12 32	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≠ 𝐵 → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
30	29	3ad2ant3 1134	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
31	30	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
32	31	imp 406	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) ∧ 𝐴 = 𝐵) → ((iEdg‘𝐺)‘𝑖) = {𝐴})
33		sseq2 4022	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = ((iEdg‘𝐺)‘𝑖) → ({𝐴, 𝐵} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖)))
34	33	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) → ({𝐴, 𝐵} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖)))
35	34	biimpa 476	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
36	35	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
37	36	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
38	24, 25, 26, 27, 32, 37, 5, 17	1pthond 30173	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → ⟨“𝑖”⟩(𝐴(PathsOn‘𝐺)𝐵)⟨“𝐴𝐵”⟩)
39		breq12 5153	. . . . . . . . . . . 12 ⊢ ((𝑓 = ⟨“𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵”⟩) → (𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ⟨“𝑖”⟩(𝐴(PathsOn‘𝐺)𝐵)⟨“𝐴𝐵”⟩))
40	39	spc2egv 3599	. . . . . . . . . . 11 ⊢ ((⟨“𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵”⟩ ∈ Word V) → (⟨“𝑖”⟩(𝐴(PathsOn‘𝐺)𝐵)⟨“𝐴𝐵”⟩ → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
41	23, 38, 40	mpsyl 68	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)
42	41	exp44 437	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (𝑖 ∈ dom (iEdg‘𝐺) → (𝑒 = ((iEdg‘𝐺)‘𝑖) → ({𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))))
43	42	rexlimdv 3151	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖) → ({𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)))
44	20, 43	sylbid 240	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (𝑒 ∈ 𝐸 → ({𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)))
45	44	rexlimdv 3151	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
46	45	3exp 1118	. . . . 5 ⊢ (𝐺 ∈ UHGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 ≠ 𝐵 → (∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))))
47	46	com34 91	. . . 4 ⊢ (𝐺 ∈ UHGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒 → (𝐴 ≠ 𝐵 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))))
48	47	3imp 1110	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒) → (𝐴 ≠ 𝐵 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
49	48	com12 32	. 2 ⊢ (𝐴 ≠ 𝐵 → ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
50	14, 49	pm2.61ine 3023	1 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  ∃wrex 3068  Vcvv 3478   ⊆ wss 3963  {csn 4631  {cpr 4633   class class class wbr 5148  dom cdm 5689  ‘cfv 6563  (class class class)co 7431  Word cword 14549  ⟨“cs1 14630  ⟨“cs2 14877  Vtxcvtx 29028  iEdgciedg 29029  Edgcedg 29079  UHGraphcuhgr 29088  PathsOncpthson 29747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-concat 14606  df-s1 14631  df-s2 14884  df-edg 29080  df-uhgr 29090  df-wlks 29632  df-wlkson 29633  df-trls 29725  df-trlson 29726  df-pths 29749  df-pthson 29751

This theorem is referenced by:  1pthon2ve  30183  cusconngr  30220