Theorem mppsthm

Description: A provable pre-statement is a theorem. (Contributed by Mario Carneiro, 18-Jul-2016)

	Ref	Expression
Hypotheses	mppsthm.j	⊢ 𝐽 = ( mPPSt ‘ 𝑇 )
	mppsthm.u	⊢ 𝑈 = ( mThm ‘ 𝑇 )
Assertion	mppsthm	⊢ 𝐽 ⊆ 𝑈

Proof

Step	Hyp	Ref	Expression
1		mppsthm.j	⊢ 𝐽 = ( mPPSt ‘ 𝑇 )
2		mppsthm.u	⊢ 𝑈 = ( mThm ‘ 𝑇 )
3		eqid	⊢ ( ( mStRed ‘ 𝑇 ) ‘ 𝑥 ) = ( ( mStRed ‘ 𝑇 ) ‘ 𝑥 )
4		eqid	⊢ ( mStRed ‘ 𝑇 ) = ( mStRed ‘ 𝑇 )
5	4 1 2	mthmi	⊢ ( ( 𝑥 ∈ 𝐽 ∧ ( ( mStRed ‘ 𝑇 ) ‘ 𝑥 ) = ( ( mStRed ‘ 𝑇 ) ‘ 𝑥 ) ) → 𝑥 ∈ 𝑈 )
6	3 5	mpan2	⊢ ( 𝑥 ∈ 𝐽 → 𝑥 ∈ 𝑈 )
7	6	ssriv	⊢ 𝐽 ⊆ 𝑈