Theorem trrelsuperreldg

Description: Concrete construction of a superclass of relation R which is a transitive relation. (Contributed by RP, 25-Dec-2019)

	Ref	Expression
Hypotheses	trrelsuperreldg.r	⊢ ( 𝜑 → Rel 𝑅 )
	trrelsuperreldg.s	⊢ ( 𝜑 → 𝑆 = ( dom 𝑅 × ran 𝑅 ) )
Assertion	trrelsuperreldg	⊢ ( 𝜑 → ( 𝑅 ⊆ 𝑆 ∧ ( 𝑆 ∘ 𝑆 ) ⊆ 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		trrelsuperreldg.r	⊢ ( 𝜑 → Rel 𝑅 )
2		trrelsuperreldg.s	⊢ ( 𝜑 → 𝑆 = ( dom 𝑅 × ran 𝑅 ) )
3		relssdmrn	⊢ ( Rel 𝑅 → 𝑅 ⊆ ( dom 𝑅 × ran 𝑅 ) )
4	1 3	syl	⊢ ( 𝜑 → 𝑅 ⊆ ( dom 𝑅 × ran 𝑅 ) )
5	4 2	sseqtrrd	⊢ ( 𝜑 → 𝑅 ⊆ 𝑆 )
6		xptrrel	⊢ ( ( dom 𝑅 × ran 𝑅 ) ∘ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( dom 𝑅 × ran 𝑅 )
7	6	a1i	⊢ ( 𝜑 → ( ( dom 𝑅 × ran 𝑅 ) ∘ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( dom 𝑅 × ran 𝑅 ) )
8	2 2	coeq12d	⊢ ( 𝜑 → ( 𝑆 ∘ 𝑆 ) = ( ( dom 𝑅 × ran 𝑅 ) ∘ ( dom 𝑅 × ran 𝑅 ) ) )
9	7 8 2	3sstr4d	⊢ ( 𝜑 → ( 𝑆 ∘ 𝑆 ) ⊆ 𝑆 )
10	5 9	jca	⊢ ( 𝜑 → ( 𝑅 ⊆ 𝑆 ∧ ( 𝑆 ∘ 𝑆 ) ⊆ 𝑆 ) )