Theorem 1pthon2v 30116

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 30092	. . . . 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 7361	. . . . . . . 8 ⊢ (𝐵 = 𝐴 → (𝐴(PathsOn‘𝐺)𝐵) = (𝐴(PathsOn‘𝐺)𝐴))
9	8	eqcoms 2737	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴(PathsOn‘𝐺)𝐵) = (𝐴(PathsOn‘𝐺)𝐴))
10	9	breqd 5106	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝))
11	10	2exbidv 1924	. . . . 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 2820	. . . . . . . . . 10 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
17		eqid 2729	. . . . . . . . . . 11 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
18	17	uhgredgiedgb 29090	. . . . . . . . . 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 14531	. . . . . . . . . . . 12 ⊢ ⟨“𝑖”⟩ ∈ Word V
22		s2cli 14806	. . . . . . . . . . . 12 ⊢ ⟨“𝐴𝐵”⟩ ∈ Word V
23	21, 22	pm3.2i 470	. . . . . . . . . . 11 ⊢ (⟨“𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵”⟩ ∈ Word V)
24		eqid 2729	. . . . . . . . . . . 12 ⊢ ⟨“𝐴𝐵”⟩ = ⟨“𝐴𝐵”⟩
25		eqid 2729	. . . . . . . . . . . 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 2936	. . . . . . . . . . . . . . . 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 3964	. . . . . . . . . . . . . . . 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 30107	. . . . . . . . . . 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 5100	. . . . . . . . . . . 12 ⊢ ((𝑓 = ⟨“𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵”⟩) → (𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ⟨“𝑖”⟩(𝐴(PathsOn‘𝐺)𝐵)⟨“𝐴𝐵”⟩))
40	39	spc2egv 3556	. . . . . . . . . . 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 3128	. . . . . . . 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 3128	. . . . . 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 3008	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 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  Vcvv 3438   ⊆ wss 3905  {csn 4579  {cpr 4581   class class class wbr 5095  dom cdm 5623  ‘cfv 6486  (class class class)co 7353  Word cword 14439  ⟨“cs1 14521  ⟨“cs2 14767  Vtxcvtx 28960  iEdgciedg 28961  Edgcedg 29011  UHGraphcuhgr 29020  PathsOncpthson 29676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-map 8762  df-pm 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-n0 12404  df-z 12491  df-uz 12755  df-fz 13430  df-fzo 13577  df-hash 14257  df-word 14440  df-concat 14497  df-s1 14522  df-s2 14774  df-edg 29012  df-uhgr 29022  df-wlks 29564  df-wlkson 29565  df-trls 29655  df-trlson 29656  df-pths 29678  df-pthson 29680

This theorem is referenced by:  1pthon2ve  30117  cusconngr  30154