Theorem pmapssbaN

Description: A weakening of pmapssat to shorten some proofs. (Contributed by NM, 7-Mar-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pmapssba.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	pmapssba.m	⊢ 𝑀 = ( pmap ‘ 𝐾 )
Assertion	pmapssbaN	⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		pmapssba.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		pmapssba.m	⊢ 𝑀 = ( pmap ‘ 𝐾 )
3		eqid	⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
4	1 3 2	pmapssat	⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) )
5	1 3	atssbase	⊢ ( Atoms ‘ 𝐾 ) ⊆ 𝐵
6	4 5	sstrdi	⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) ⊆ 𝐵 )