Theorem 1pthon2v 30125

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 1132	. . . . . 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 30101	. . . . 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 7349	. . . . . . . 8 ⊢ (𝐵 = 𝐴 → (𝐴(PathsOn‘𝐺)𝐵) = (𝐴(PathsOn‘𝐺)𝐴))
9	8	eqcoms 2739	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴(PathsOn‘𝐺)𝐵) = (𝐴(PathsOn‘𝐺)𝐴))
10	9	breqd 5097	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝))
11	10	2exbidv 1925	. . . . 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 2823	. . . . . . . . . 10 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
17		eqid 2731	. . . . . . . . . . 11 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
18	17	uhgredgiedgb 29099	. . . . . . . . . 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 1133	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (𝑒 ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖)))
21		s1cli 14508	. . . . . . . . . . . 12 ⊢ ⟨“𝑖”⟩ ∈ Word V
22		s2cli 14782	. . . . . . . . . . . 12 ⊢ ⟨“𝐴𝐵”⟩ ∈ Word V
23	21, 22	pm3.2i 470	. . . . . . . . . . 11 ⊢ (⟨“𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵”⟩ ∈ Word V)
24		eqid 2731	. . . . . . . . . . . 12 ⊢ ⟨“𝐴𝐵”⟩ = ⟨“𝐴𝐵”⟩
25		eqid 2731	. . . . . . . . . . . 12 ⊢ ⟨“𝑖”⟩ = ⟨“𝑖”⟩
26		simpl2l 1227	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → 𝐴 ∈ 𝑉)
27		simpl2r 1228	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → 𝐵 ∈ 𝑉)
28		eqneqall 2939	. . . . . . . . . . . . . . . 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 1135	. . . . . . . . . . . . . 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 3956	. . . . . . . . . . . . . . . 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 30116	. . . . . . . . . . 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 5091	. . . . . . . . . . . 12 ⊢ ((𝑓 = ⟨“𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵”⟩) → (𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ⟨“𝑖”⟩(𝐴(PathsOn‘𝐺)𝐵)⟨“𝐴𝐵”⟩))
40	39	spc2egv 3549	. . . . . . . . . . 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 3131	. . . . . . . 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 3131	. . . . . 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 1119	. . . . 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 3011	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 1541  ∃wex 1780   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056  Vcvv 3436   ⊆ wss 3897  {csn 4571  {cpr 4573   class class class wbr 5086  dom cdm 5611  ‘cfv 6476  (class class class)co 7341  Word cword 14415  ⟨“cs1 14498  ⟨“cs2 14743  Vtxcvtx 28969  iEdgciedg 28970  Edgcedg 29020  UHGraphcuhgr 29029  PathsOncpthson 29685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

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 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-er 8617  df-map 8747  df-pm 8748  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-card 9827  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-2 12183  df-n0 12377  df-z 12464  df-uz 12728  df-fz 13403  df-fzo 13550  df-hash 14233  df-word 14416  df-concat 14473  df-s1 14499  df-s2 14750  df-edg 29021  df-uhgr 29031  df-wlks 29573  df-wlkson 29574  df-trls 29664  df-trlson 29665  df-pths 29687  df-pthson 29689

This theorem is referenced by:  1pthon2ve  30126  cusconngr  30163