Theorem spthdifv

Description: The vertices of a simple path are distinct, so the vertex function is one-to-one. (Contributed by Alexander van der Vekens, 26-Jan-2018) (Revised by AV, 5-Jun-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Assertion	spthdifv	$$⊢F\mathrm{SPaths}\left(G\right)P\to P:\left(0\dots \left|F\right|\right)\underset{1-1}{⟶}\mathrm{Vtx}\left(G\right)$$

Proof

Step	Hyp	Ref	Expression
1		isspth	$$⊢F\mathrm{SPaths}\left(G\right)P\leftrightarrow \left(F\mathrm{Trails}\left(G\right)P\wedge \mathrm{Fun}{P}^{-1}\right)$$
2		trliswlk	$$⊢F\mathrm{Trails}\left(G\right)P\to F\mathrm{Walks}\left(G\right)P$$
3		eqid	$$⊢\mathrm{Vtx}\left(G\right)=\mathrm{Vtx}\left(G\right)$$
4	3	wlkp	$$⊢F\mathrm{Walks}\left(G\right)P\to P:\left(0\dots \left|F\right|\right)⟶\mathrm{Vtx}\left(G\right)$$
5		df-f1	$$⊢P:\left(0\dots \left|F\right|\right)\underset{1-1}{⟶}\mathrm{Vtx}\left(G\right)\leftrightarrow \left(P:\left(0\dots \left|F\right|\right)⟶\mathrm{Vtx}\left(G\right)\wedge \mathrm{Fun}{P}^{-1}\right)$$
6	5	simplbi2	$$⊢P:\left(0\dots \left|F\right|\right)⟶\mathrm{Vtx}\left(G\right)\to \left(\mathrm{Fun}{P}^{-1}\to P:\left(0\dots \left|F\right|\right)\underset{1-1}{⟶}\mathrm{Vtx}\left(G\right)\right)$$
7	2 4 6	3syl	$$⊢F\mathrm{Trails}\left(G\right)P\to \left(\mathrm{Fun}{P}^{-1}\to P:\left(0\dots \left|F\right|\right)\underset{1-1}{⟶}\mathrm{Vtx}\left(G\right)\right)$$
8	7	imp	$$⊢\left(F\mathrm{Trails}\left(G\right)P\wedge \mathrm{Fun}{P}^{-1}\right)\to P:\left(0\dots \left|F\right|\right)\underset{1-1}{⟶}\mathrm{Vtx}\left(G\right)$$
9	1 8	sylbi	$$⊢F\mathrm{SPaths}\left(G\right)P\to P:\left(0\dots \left|F\right|\right)\underset{1-1}{⟶}\mathrm{Vtx}\left(G\right)$$