usgr2pth - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > usgr2pth		Structured version Visualization version GIF version

Theorem usgr2pth 30122

Description: In a simple graph, there is a path of length 2 iff there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.) (Revised by AV, 5-Jun-2021.) (Proof shortened by AV, 31-Oct-2021.)

**Hypotheses**
Ref	Expression
usgr2pthlem.v	⊢ 𝑉 = (Vtx‘𝐺)
usgr2pthlem.i	⊢ 𝐼 = (iEdg‘𝐺)

**Assertion**
Ref	Expression
usgr2pth	⊢ (𝐺 ∈ USGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})))))

Distinct variable groups: 𝑥,𝐹,𝑦,𝑧 𝑥,𝐺,𝑦,𝑧 𝑥,𝐼,𝑦,𝑧 𝑥,𝑃,𝑦,𝑧 𝑥,𝑉,𝑦,𝑧

Proof of Theorem usgr2pth Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgr2pthspth 30120	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Paths‘𝐺)𝑃 ↔ 𝐹(SPaths‘𝐺)𝑃))
2		usgrupgr 29544	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
3	2	adantr 485	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝐺 ∈ UPGraph)
4		isspth 30080	. . . . . . . . 9 ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃))
5	4	a1i 11	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃)))
6		usgr2pthlem.v	. . . . . . . . . . 11 ⊢ 𝑉 = (Vtx‘𝐺)
7		usgr2pthlem.i	. . . . . . . . . . 11 ⊢ 𝐼 = (iEdg‘𝐺)
8	6, 7	upgrf1istrl 30060	. . . . . . . . . 10 ⊢ (𝐺 ∈ UPGraph → (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
9	8	anbi1d 642	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃) ↔ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃)))
10		oveq2 7418	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
11		f1eq2 6770	. . . . . . . . . . . . . . . . 17 ⊢ ((0..^(♯‘𝐹)) = (0..^2) → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ↔ 𝐹:(0..^2)–1-1→dom 𝐼))
12	10, 11	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) = 2 → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ↔ 𝐹:(0..^2)–1-1→dom 𝐼))
13	12	biimpd 232	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) = 2 → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → 𝐹:(0..^2)–1-1→dom 𝐼))
14	13	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → 𝐹:(0..^2)–1-1→dom 𝐼))
15	14	com12 33	. . . . . . . . . . . . 13 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝐹:(0..^2)–1-1→dom 𝐼))
16	15	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝐹:(0..^2)–1-1→dom 𝐼))
17	16	ad2antrl 740	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃)) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝐹:(0..^2)–1-1→dom 𝐼))
18		oveq2 7418	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) = 2 → (0...(♯‘𝐹)) = (0...2))
19	18	feq2d 6689	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) = 2 → (𝑃:(0...(♯‘𝐹))⟶𝑉 ↔ 𝑃:(0...2)⟶𝑉))
20		df-f1 6541	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃:(0...2)–1-1→𝑉 ↔ (𝑃:(0...2)⟶𝑉 ∧ Fun ^◡𝑃))
21	20	simplbi2 505	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃:(0...2)⟶𝑉 → (Fun ^◡𝑃 → 𝑃:(0...2)–1-1→𝑉))
22	21	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) = 2 → (𝑃:(0...2)⟶𝑉 → (Fun ^◡𝑃 → 𝑃:(0...2)–1-1→𝑉)))
23	19, 22	sylbid 243	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) = 2 → (𝑃:(0...(♯‘𝐹))⟶𝑉 → (Fun ^◡𝑃 → 𝑃:(0...2)–1-1→𝑉)))
24	23	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝑃:(0...(♯‘𝐹))⟶𝑉 → (Fun ^◡𝑃 → 𝑃:(0...2)–1-1→𝑉)))
25	24	com3l 90	. . . . . . . . . . . . . 14 ⊢ (𝑃:(0...(♯‘𝐹))⟶𝑉 → (Fun ^◡𝑃 → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝑃:(0...2)–1-1→𝑉)))
26	25	3ad2ant2 1152	. . . . . . . . . . . . 13 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (Fun ^◡𝑃 → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝑃:(0...2)–1-1→𝑉)))
27	26	imp 411	. . . . . . . . . . . 12 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝑃:(0...2)–1-1→𝑉))
28	27	adantl 486	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃)) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝑃:(0...2)–1-1→𝑉))
29	6, 7	usgr2pthlem 30121	. . . . . . . . . . . 12 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))
30	29	ad2antrl 740	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃)) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))
31	17, 28, 30	3jcad 1147	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃)) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})))))
32	31	ex 417	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → (((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))))
33	9, 32	sylbid 243	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))))
34	5, 33	sylbid 243	. . . . . . 7 ⊢ (𝐺 ∈ UPGraph → (𝐹(SPaths‘𝐺)𝑃 → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))))
35	34	com23 87	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(SPaths‘𝐺)𝑃 → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))))
36	3, 35	mpcom 39	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(SPaths‘𝐺)𝑃 → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})))))
37	1, 36	sylbid 243	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Paths‘𝐺)𝑃 → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})))))
38	37	ex 417	. . 3 ⊢ (𝐺 ∈ USGraph → ((♯‘𝐹) = 2 → (𝐹(Paths‘𝐺)𝑃 → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))))
39	38	impcomd 416	. 2 ⊢ (𝐺 ∈ USGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})))))
40		2nn0 12525	. . . . . 6 ⊢ 2 ∈ ℕ₀
41		f1f 6774	. . . . . 6 ⊢ (𝐹:(0..^2)–1-1→dom 𝐼 → 𝐹:(0..^2)⟶dom 𝐼)
42		fnfzo0hash 14492	. . . . . 6 ⊢ ((2 ∈ ℕ₀ ∧ 𝐹:(0..^2)⟶dom 𝐼) → (♯‘𝐹) = 2)
43	40, 41, 42	sylancr 598	. . . . 5 ⊢ (𝐹:(0..^2)–1-1→dom 𝐼 → (♯‘𝐹) = 2)
44		oveq2 7418	. . . . . . . . . . . . . . . . . 18 ⊢ (2 = (♯‘𝐹) → (0..^2) = (0..^(♯‘𝐹)))
45	44	eqcoms 2771	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) = 2 → (0..^2) = (0..^(♯‘𝐹)))
46		f1eq2 6770	. . . . . . . . . . . . . . . . 17 ⊢ ((0..^2) = (0..^(♯‘𝐹)) → (𝐹:(0..^2)–1-1→dom 𝐼 ↔ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼))
47	45, 46	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) = 2 → (𝐹:(0..^2)–1-1→dom 𝐼 ↔ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼))
48	47	biimpd 232	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) = 2 → (𝐹:(0..^2)–1-1→dom 𝐼 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼))
49	48	imp 411	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
50	49	adantr 485	. . . . . . . . . . . . 13 ⊢ ((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
51	50	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
52		f1f 6774	. . . . . . . . . . . . . . 15 ⊢ (𝑃:(0...2)–1-1→𝑉 → 𝑃:(0...2)⟶𝑉)
53		oveq2 7418	. . . . . . . . . . . . . . . . . 18 ⊢ (2 = (♯‘𝐹) → (0...2) = (0...(♯‘𝐹)))
54	53	eqcoms 2771	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) = 2 → (0...2) = (0...(♯‘𝐹)))
55	54	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) → (0...2) = (0...(♯‘𝐹)))
56	55	feq2d 6689	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) → (𝑃:(0...2)⟶𝑉 ↔ 𝑃:(0...(♯‘𝐹))⟶𝑉))
57	52, 56	imbitrid 247	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) → (𝑃:(0...2)–1-1→𝑉 → 𝑃:(0...(♯‘𝐹))⟶𝑉))
58	57	imp 411	. . . . . . . . . . . . 13 ⊢ ((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) → 𝑃:(0...(♯‘𝐹))⟶𝑉)
59	58	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) → 𝑃:(0...(♯‘𝐹))⟶𝑉)
60		eqcom 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃‘0) = 𝑥 ↔ 𝑥 = (𝑃‘0))
61	60	biimpi 219	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃‘0) = 𝑥 → 𝑥 = (𝑃‘0))
62	61	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → 𝑥 = (𝑃‘0))
63		eqcom 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃‘1) = 𝑦 ↔ 𝑦 = (𝑃‘1))
64	63	biimpi 219	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃‘1) = 𝑦 → 𝑦 = (𝑃‘1))
65	64	3ad2ant2 1152	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → 𝑦 = (𝑃‘1))
66	62, 65	preq12d 4707	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → {𝑥, 𝑦} = {(𝑃‘0), (𝑃‘1)})
67	66	eqeq2d 2774	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ↔ (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
68	67	biimpcd 252	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} → (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
69	68	adantr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}) → (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
70	69	impcom 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})) → (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)})
71		eqcom 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃‘2) = 𝑧 ↔ 𝑧 = (𝑃‘2))
72	71	biimpi 219	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃‘2) = 𝑧 → 𝑧 = (𝑃‘2))
73	72	3ad2ant3 1153	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → 𝑧 = (𝑃‘2))
74	65, 73	preq12d 4707	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → {𝑦, 𝑧} = {(𝑃‘1), (𝑃‘2)})
75	74	eqeq2d 2774	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → ((𝐼‘(𝐹‘1)) = {𝑦, 𝑧} ↔ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
76	75	biimpcd 252	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐼‘(𝐹‘1)) = {𝑦, 𝑧} → (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
77	76	adantl 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}) → (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
78	77	impcom 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})) → (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})
79	70, 78	jca 520	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})) → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
80	79	rexlimivw 3162	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})) → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
81	80	rexlimivw 3162	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})) → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
82	81	rexlimivw 3162	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})) → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
83	82	a1i13 28	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) = 2 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})) → (𝐺 ∈ USGraph → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
84		fzo0to2pr 13784	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0..^2) = {0, 1}
85	10, 84	eqtrdi 2814	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = {0, 1})
86	85	raleqdv 3323	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) = 2 → (∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ∀𝑖 ∈ {0, 1} (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
87		2wlklem 30024	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑖 ∈ {0, 1} (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
88	86, 87	bitrdi 290	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) = 2 → (∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
89	88	imbi2d 343	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) = 2 → ((𝐺 ∈ USGraph → ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ↔ (𝐺 ∈ USGraph → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
90	83, 89	sylibrd 262	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) = 2 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})) → (𝐺 ∈ USGraph → ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
91	90	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ ((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})) → (𝐺 ∈ USGraph → ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
92	91	imp 411	. . . . . . . . . . . . 13 ⊢ (((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) → (𝐺 ∈ USGraph → ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
93	92	imp 411	. . . . . . . . . . . 12 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) → ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})
94	51, 59, 93	3jca 1146	. . . . . . . . . . 11 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
95	20	simprbi 502	. . . . . . . . . . . . 13 ⊢ (𝑃:(0...2)–1-1→𝑉 → Fun ^◡𝑃)
96	95	adantl 486	. . . . . . . . . . . 12 ⊢ ((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) → Fun ^◡𝑃)
97	96	ad2antrr 738	. . . . . . . . . . 11 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) → Fun ^◡𝑃)
98	94, 97	jca 520	. . . . . . . . . 10 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) → ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃))
99	5, 9	bitrd 282	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ UPGraph → (𝐹(SPaths‘𝐺)𝑃 ↔ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃)))
100	2, 99	syl 18	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → (𝐹(SPaths‘𝐺)𝑃 ↔ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃)))
101	100	adantl 486	. . . . . . . . . 10 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) → (𝐹(SPaths‘𝐺)𝑃 ↔ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃)))
102	98, 101	mpbird 260	. . . . . . . . 9 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) → 𝐹(SPaths‘𝐺)𝑃)
103		simpr 489	. . . . . . . . . 10 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) → 𝐺 ∈ USGraph)
104		simp-4l 794	. . . . . . . . . 10 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) → (♯‘𝐹) = 2)
105	103, 104, 1	syl2anc 595	. . . . . . . . 9 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) → (𝐹(Paths‘𝐺)𝑃 ↔ 𝐹(SPaths‘𝐺)𝑃))
106	102, 105	mpbird 260	. . . . . . . 8 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) → 𝐹(Paths‘𝐺)𝑃)
107	106, 104	jca 520	. . . . . . 7 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) → (𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2))
108	107	ex 417	. . . . . 6 ⊢ (((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) → (𝐺 ∈ USGraph → (𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2)))
109	108	exp41 439	. . . . 5 ⊢ ((♯‘𝐹) = 2 → (𝐹:(0..^2)–1-1→dom 𝐼 → (𝑃:(0...2)–1-1→𝑉 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})) → (𝐺 ∈ USGraph → (𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2))))))
110	43, 109	mpcom 39	. . . 4 ⊢ (𝐹:(0..^2)–1-1→dom 𝐼 → (𝑃:(0...2)–1-1→𝑉 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})) → (𝐺 ∈ USGraph → (𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2)))))
111	110	3imp 1128	. . 3 ⊢ ((𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) → (𝐺 ∈ USGraph → (𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2)))
112	111	com12 33	. 2 ⊢ (𝐺 ∈ USGraph → ((𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) → (𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2)))
113	39, 112	impbid 215	1 ⊢ (𝐺 ∈ USGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 ∖ cdif 3902 {csn 4589 {cpr 4591 class class class wbr 5109 ^◡ccnv 5660 dom cdm 5661 Fun wfun 6530 ⟶wf 6532 –1-1→wf1 6533 ‘cfv 6536 (class class class)co 7410 0cc0 11104 1c1 11105 + caddc 11107 2c2 12299 ℕ₀cn0 12508 ...cfz 13539 ..^cfzo 13687 ♯chash 14371 Vtxcvtx 29355 iEdgciedg 29356 UPGraphcupgr 29439 USGraphcusgr 29508 Trailsctrls 30047 Pathscpths 30068 SPathscspths 30069

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 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 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-er 8690 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9892 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-nn 12238 df-2 12307 df-3 12308 df-n0 12509 df-xnn0 12582 df-z 12596 df-uz 12867 df-fz 13540 df-fzo 13688 df-hash 14372 df-word 14556 df-concat 14613 df-s1 14639 df-s2 14890 df-s3 14891 df-edg 29407 df-uhgr 29417 df-upgr 29441 df-umgr 29442 df-uspgr 29509 df-usgr 29510 df-wlks 29958 df-wlkson 29959 df-trls 30049 df-trlson 30050 df-pths 30072 df-spths 30073 df-pthson 30074 df-spthson 30075

This theorem is used by: usgr2pth0 30123

W3C validator