Theorem trkgitv

Description: The congruence relation in a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017)

	Ref	Expression
Hypothesis	trkgstr.w	⊢ 𝑊 = { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ , ⟨ ( Itv ‘ ndx ) , 𝐼 ⟩ }
Assertion	trkgitv	⊢ ( 𝐼 ∈ 𝑉 → 𝐼 = ( Itv ‘ 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		trkgstr.w	⊢ 𝑊 = { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ , ⟨ ( Itv ‘ ndx ) , 𝐼 ⟩ }
2	1	trkgstr	⊢ 𝑊 Struct ⟨ 1 , ; 1 6 ⟩
3		itvid	⊢ Itv = Slot ( Itv ‘ ndx )
4		snsstp3	⊢ { ⟨ ( Itv ‘ ndx ) , 𝐼 ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ , ⟨ ( Itv ‘ ndx ) , 𝐼 ⟩ }
5	4 1	sseqtrri	⊢ { ⟨ ( Itv ‘ ndx ) , 𝐼 ⟩ } ⊆ 𝑊
6	2 3 5	strfv	⊢ ( 𝐼 ∈ 𝑉 → 𝐼 = ( Itv ‘ 𝑊 ) )