2spthd

Description: A simple path of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 1-Feb-2018) (Revised by AV, 24-Jan-2021) (Revised by AV, 24-Mar-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Hypotheses	2wlkd.p	$⊢ P = ⟨“ A B C ”⟩$
	2wlkd.f	$⊢ F = ⟨“ J K ”⟩$
	2wlkd.s	$⊢ φ \to (A \in V \land B \in V \land C \in V)$
	2wlkd.n	$⊢ φ \to (A \neq B \land B \neq C)$
	2wlkd.e	$⊢ φ \to (\{A, B\} \subseteq I (J) \land \{B, C\} \subseteq I (K))$
	2wlkd.v	$⊢ V = Vtx (G)$
	2wlkd.i	$⊢ I = iEdg (G)$
	2trld.n	$⊢ φ \to J \neq K$
	2spthd.n	$⊢ φ \to A \neq C$
Assertion	2spthd	$⊢ φ \to F SPaths (G) P$

Proof

Step	Hyp	Ref	Expression
1		2wlkd.p	$⊢ P = ⟨“ A B C ”⟩$
2		2wlkd.f	$⊢ F = ⟨“ J K ”⟩$
3		2wlkd.s	$⊢ φ \to (A \in V \land B \in V \land C \in V)$
4		2wlkd.n	$⊢ φ \to (A \neq B \land B \neq C)$
5		2wlkd.e	$⊢ φ \to (\{A, B\} \subseteq I (J) \land \{B, C\} \subseteq I (K))$
6		2wlkd.v	$⊢ V = Vtx (G)$
7		2wlkd.i	$⊢ I = iEdg (G)$
8		2trld.n	$⊢ φ \to J \neq K$
9		2spthd.n	$⊢ φ \to A \neq C$
10	1 2 3 4 5 6 7 8	2trld	$⊢ φ \to F Trails (G) P$
11		3anan32	$⊢ (A \neq B \land A \neq C \land B \neq C) \leftrightarrow ((A \neq B \land B \neq C) \land A \neq C)$
12	4 9 11	sylanbrc	$⊢ φ \to (A \neq B \land A \neq C \land B \neq C)$
13		funcnvs3	$⊢ ((A \in V \land B \in V \land C \in V) \land (A \neq B \land A \neq C \land B \neq C)) \to Fun {⟨“ A B C ”⟩}^{-1}$
14	3 12 13	syl2anc	$⊢ φ \to Fun {⟨“ A B C ”⟩}^{-1}$
15	1	a1i	$⊢ φ \to P = ⟨“ A B C ”⟩$
16	15	cnveqd	$⊢ φ \to P^{-1} = {⟨“ A B C ”⟩}^{-1}$
17	16	funeqd	$⊢ φ \to (Fun P^{-1} \leftrightarrow Fun {⟨“ A B C ”⟩}^{-1})$
18	14 17	mpbird	$⊢ φ \to Fun P^{-1}$
19		isspth	$⊢ F SPaths (G) P \leftrightarrow (F Trails (G) P \land Fun P^{-1})$
20	10 18 19	sylanbrc	$⊢ φ \to F SPaths (G) P$

Metamath Proof Explorer

Theorem 2spthd

Proof